digitalassets.lib.berkeley.edu · 1 Abstract Pointwise Ergodic Theorems for Nonconventional L1...

Pointwise Ergodic Theorems for Nonconventional L1 Averages

by

Patrick LaVictoire

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Mathematics

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor F. Michael Christ, ChairProfessor Daniel Tataru

Assistant Professor Noureddine El Karoui

Spring 2010


Copyright 2010by

Patrick LaVictoire

1

Abstract


by

Patrick LaVictoire

Doctor of Philosophy in Mathematics

University of California, Berkeley

Professor F. Michael Christ, Chair

A topic of classical interest in ergodic theory is extending Birkhoff’s Pointwise ErgodicTheorem to various classes of nonconventional ergodic averages. Previous methods had onlyestablished the pointwise behavior of many such averages when applied to functions in Lp

with p > 1, leaving open the important case of L1.

In this thesis, I adapt and extend the techniques of Urban and Zienkiewicz [30] and Buczolichand Mauldin [11], whose recent L1 results have renewed interest in the subject, in order toprove several new results on the pointwise convergence of nonconventional averages of L1

functions.

The first result (Theorem 1.3) proves that averages along certain Bernoulli random sequences(generated by independent 0, 1-valued random variables) satisfy a pointwise ergodic the-orem in L1, with probability 1. The second (Theorem 1.4) proves that averages along nd

for d ≥ 2, or along the sequence of primes, do not satisfy a pointwise ergodic theorem inL1. (The result for n2 was first shown in [11]; the others are new.) The third (Corollary1.9) shows that, given a Fourier decay condition on a sequence of probability measures onZ, some subsequence of the corresponding nonconventional averages will satisfy a pointwiseergodic theorem in L1.

i

Contents

1 Introduction 11.1 Context: Ergodic Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Random Subsequences of Density 0 . . . . . . . . . . . . . . . . . . . . . . . 41.3 Universally L1-Bad Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . 51.4 Sequences of Convolution Operators . . . . . . . . . . . . . . . . . . . . . . . 61.5 The Banach Principle and the Conze Transference Principle . . . . . . . . . 71.6 Two Recurring Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 L2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.8 L1 Theory: Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Sequences of Convolution Operators 162.1 Approximate Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Weighted Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Non-Uniform Fourier Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Random Subsequences of Density 0 233.1 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Calderon-Zygmund Argument . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Probabilistic Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Universally L1-Bad Arithmetic Sequences 324.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Outline of the Argument for the Squares . . . . . . . . . . . . . . . . . . . . 334.3 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 The Inductive Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5 Periodic Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.6 Defining fL+1

K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.7 Restricting a Family to (−Λq)

γ . . . . . . . . . . . . . . . . . . . . . . . . . 464.8 Completion of the Inductive Step . . . . . . . . . . . . . . . . . . . . . . . . 484.9 Notes on the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

ii

Bibliography 54

iii

Acknowledgments

The author thanks his thesis advisor, M. Christ, for years of invaluable ideas, advice, heuris-tics, and direction; J. Rosenblatt and M. Wierdl for many conversations, suggestions andencouragement, particularly on the results in Sections 2 and 4 (respectively); many othermathematicians whose correspondence and support have been greatly helpful at variouspoints, including C. Demeter, C. Thiele, M. Lacey, B. Stovall, and K. Datchev; the authorsF. Dostoyevsky, F. Nietzsche and D. Hofstadter, among others; his extraordinary friends andfamily; and you, for reading this.

1

Chapter 1

Introduction

1.1 Context: Ergodic Theorems

The starting point for this thesis is the Birkhoff pointwise ergodic theorem, as we will bediscussing various extensions of this to other weighted ergodic averages.

Definition A dynamical system (X,F ,m, T ) consists of a topological space X with σ-algebra F , a probability measure m on (X,F), and a measure-preserving transformationT : X → X; that is, m(T−1A) = m(A) ∀A ∈ F . We will usually suppress F in all thatfollows.

Definition A dynamical system (X,F ,m, T ) is ergodic if it is non-atomic (i.e. ∀x ∈X,m(x) = 0) and if every T -invariant set A = T−1A ∈ F has measure 0 or 1.

Example Let X = T = R/Z, with Lebesgue measure and the measure-preserving transfor-mation Tαx = x+ α (mod 1). If α /∈ Q, the Fourier transform reveals that any Tα-invariantfunction f = f Tα ∈ L2(T) must be constant a.e, and therefore the dynamical system isergodic. If α = p

q, however, the set A = x ∈ T : (qx mod 1) ∈ [0, 1/2) is a nontrivial

Tα-invariant set.

Example Let X = ZN , with the counting measure (rescaled by 1/N to be a probabilitymeasure) and the measure-preserving transformation Tk = k + 1 (mod N). Note that if Nis prime, this dynamical system has no nontrivial T -invariant sets. The notion of an ergodicdynamical system is defined in order to exclude this example, because as we shall see, thereis behavior shared by all ergodic dynamical systems but not by such systems as this.

Example Let Let X = T = R/Z, with Lebesgue measure and the “doubling map” Tx = 2x(mod 1). While this map is two-to-one (however, note Remark 5 below), it indeed hasthe property that m(T−1A) = m(A) ∀A ∈ F , and is a well-studied ergodic transformation(again, consider the Fourier transform of the characteristic function of an invariant set).

CHAPTER 1. INTRODUCTION 2

There are two very different classical versions of the ergodic theorem, proved by Von Neu-mann and Birkhoff in the 1930s:

Theorem 1.1 (Von Neumann Mean Ergodic Theorem). Let (X,F ,m, T ) be a dynamical

system, and f ∈ L2(X,m). Then the averages ANf(x) :=1

N

N∑k=1

f(T kx) converge in the L2

norm. If (X,F ,m, T ) is ergodic, then the limit of these averages is the constant function∫X

f dm.

Theorem 1.2 (Birkhoff Pointwise Ergodic Theorem). Let (X,F ,m, T ) be a dynamical sys-

tem, and f ∈ L1(X,m). Then the averages ANf(x) :=1

N

N∑k=1

f(T kx) converge for almost

every x ∈ X (by the measure m). If (X,F ,m, T ) is ergodic, then the a.e. limit of these

averages is the constant

∫X

f dm.

We will be considering extensions of ergodic theorems to sequences of “nonconventional”weighted ergodic averages. For any sequence1 of complex numbers aN,jN∈N,j∈Z such that∑

j∈Z |aN,j| < ∞ for all N , we may define for any dynamical system (X,F ,m, T ) and any

f ∈ L1(X,m) the weighted ergodic averages

ANf(x) :=∑j∈Z

aN,jf(T jx), (1.1)

and we are interested in whether this sequence of weighted averages converges (in the meanor a.e.) for all dynamical systems and all functions in a given Lp class.

Remarks:

1. The mean or pointwise limit for a sequence of weighted averages need not equal

∫X

f dm

if it exists. We will only be concerned with whether such a limit exists in the first place.

2. A pointwise ergodic theorem for a sequence of weighted ergodic averages easily impliesthe corresponding mean ergodic theorem, via the Dominated Convergence Theorem; amean ergodic theorem does not imply a pointwise ergodic theorem.

3. A mean ergodic theorem is equivalent to the same statement with L2 replaced by Lp,for any 1 ≤ p <∞.

1For such general arrays, it is an abuse of notation to call the objects (1.1) “averages”; but in all thatfollows (save for a remark at the end of Chapter 2), we will be concerned only with averages correspondingto probability measures on Z.


4. A pointwise ergodic theorem for Lp functions implies one for Lq functions if and onlyif p ≤ q. For example, Bellow [4] constructed sequences for any p > 1 so that thecorresponding weighted averages have pointwise ergodic theorems for Lp but not forany Lq with q < p.

5. It may seem illegitimate to allow negative powers of T in weighted ergodic averages.However, for our purposes, we do not lose any generality by restricting ourselves toinvertible transformations T , as the sequence f T n : n ≥ 0 is always equal in jointdistribution to the corresponding sequence for some invertible transformation. See forexample [22], Theorem 4.8.

A bit of notation will be useful for several of the results in this thesis. One important typeof nonconventional ergodic average concerns the average along a sequence of integers nk;for any (X,F ,m, T ) and any f ∈ L1(X,m), we can define the subsequence averages

AnkN f(x) :=

1

N

N∑k=1

f(T nkx). (1.2)

Definition We say that nk is universally Lp-good if for every dynamical system (X,F ,m, T )

and every f ∈ Lp(X,m), limN→∞

AnkN f(x) exists for almost every x ∈ X. We say that nk

is universally Lp-bad if for every ergodic dynamical system (X,F ,m, T ), there exists an

f ∈ Lp(X,m) such that the sequence AnkN f(x)∞N=1 diverges on a set of positive measurein X.

It is easy to construct sequences which are universally Lp-bad; for instance, nk = 2k (orindeed, any lacunary sequence of natural numbers) is universally L∞-bad (see [23] or [1]).On the other hand, Bourgain proved that certain arithmetic and other sequences were uni-versally L2-good, in particular sequences of polynomial values nk = bp(k)c [9], the sequenceof prime numbers [7], and (with probability 1) certain Bernoulli random sequences of density0 [8].

The L2 theory has turned out to correspond well to the study of related exponential sums(see Section 1.7); a nearly complete characterization [6] of the integer parts of subpolyno-mial functions in Hardy fields shows that e.g. nk = k2 + blog kc is universally L2-bad butnk = k2 + bk log kc is universally L2-good.

The main topics of this thesis are L1 pointwise ergodic theorems for certain sequences ofnonconventional ergodic averages. In each case, an L2 pointwise ergodic theorem (and thusalso a mean ergodic theorem) had been known, and even extended to Lp for any p > 1,but the behavior for averages of L1 functions was unknown– indeed, the analysis of the cor-responding exponential sums does not produce good enough bounds to handle L1. These


L1 pointwise ergodic theorems are instead investigated (see Section 1.5) by considering thevalidity of weak (1,1) maximal inequalities for corresponding sequences of convolution oper-ators on `1(Z).

A few words on the organization of this thesis: we will begin by stating our results inSections 1.2-1.4, then explain a few key background results which will be necessary for us(Sections 1.5-1.6). We will then briefly explain the reasoning behind the various L2 ergodictheorems on which we rely (Section 1.7), before explaining the heuristics behind our positiveresults (Section 1.8). We will then prove our results in Chapters 2 through 4.

1.2 Random Subsequences of Density 0

The first main theorem is a positive result concerning ergodic averages along a sparserandom subsequence. This was one of the first published results proving a genuinely sparsesequence to be universally L1-good.

Universally L1-good sequences of upper density 0 had been known to exist since the 1980s:Bellow and Losert [3] showed that for any F : N → R+, there exists a universally L1-goodblock sequence nk with nk ≥ F (k). These examples were “block sequences”, consistingof large intervals of consecutive integers separated by wide gaps. The behavior of ergodicaverages along block sequences is so similar to that of the standard ergodic averages thatthe standard maximal ergodic theorem could be applied to them (see Section 1.8). To dis-tinguish these block sequences from other sequences of interest, we bring in the notion ofupper Banach density:

Definition A sequence of positive integers nk has upper Banach density c if

limm→∞

supN

|nk ∈ [N,N +m)|m

= c.

Block sequences with arbitrarily large block lengths have upper Banach density 1 (the se-quences in [3] are all of this sort). The first example of a universally L1-good sequence withupper Banach density 0 was published in 2007 by Buczolich [10], via a delicate arithmeticconstruction. Urban and Zienkiewicz [30] subsequently proved that the power sequence bkacfor 1 < a < 1 + 1

1000is universally L1-good.

A class of sparse random sequences had been of prior interest in this subject. These se-quences are generated as follows: given a decreasing sequence of probabilities τj : j ∈ N,let ξj : j ∈ N be independent random variables on a probability space Ω with P(ξj =1) = τj, P(ξj = 0) = 1 − τj. Then for each ω ∈ Ω, define a random sequence by taking theset n : ξn(ω) = 1 in increasing order. For α > 0 and τj = O(j−α), these sequences have


Banach density 0 with probability 1; see Proposition 3.5.

Bourgain [8] had proved that for p > 1, if τj ≥ j−1(log log j)(p−1)−1+ε then the randomsequence was universally Lp-good with probability 1. However, as discussed below in Sec-tion 1.8, even the exceptionally good bounds on the associated exponential sums did notlead to an L1 result by this method.

We applied a construction of [30] (inspired by [14] and [18]) to these random sequencesand thereby proved the following L1 ergodic theorem:

Theorem 1.3. Let 0 < α < 1/2, and let ξn be independent 0, 1-valued random variableson Ω with P(ξn = 1) = n−α. Then there exists a set Ω′ ⊂ Ω of probability 1 such that forevery ω ∈ Ω′, for every dynamical system (X,F ,m, T ) and every f ∈ L1(X,m), the averages

A(ω)N f(x) = Nα−1

N∑n=1

ξn(ω)f(T nx) (1.3)

converge for a.e. x ∈ X. Thus, with probability 1, n : ξn(ω) = 1 is universally L1-good.

With probability 1, these sequences grow like k1/(1−γ), so Theorem 1.3 applies to randomsequences nearly as sparse as the sequence of squares. Also, the argument works for anysequence of independent 0 ≤ ξj ≤ 1 such that Eξj are nonincreasing and Eξj & j−α for someα < 1/2.

These results are proved in Chapter 3.

1.3 Universally L1-Bad Arithmetic Sequences

The second main theorem is a negative result concerning the sequence of dth powersnk = kd, the sequence of primes, and other sequences with similar Diophantine properties tothese.

This result follows on Buczolich and Mauldin’s proof [11] [12] that k2 is in fact universallyL1-bad, using an iterative construction that exploits the regularities of the squares in residueclasses.

We adapted and extended that construction, so that it applied to the sequences mentionedabove, and also so that it proved an even stronger property than universal L1-badness forthese sequences:

Definition We say that nk is persistently universally Lp-bad if for every ergodic dynamicalsystem (X,F , µ, τ) and every infinite S ⊂ N, there exists an f ∈ Lp(X,µ) such that the


sequence ANf(x)N∈S (N taken in increasing order) diverges on a set of positive measurein X.

Theorem 1.4. Let nk = kd for some d > 1, or nk = the kth prime number. Given anyC > 0 and any infinite set S ⊂ N, there exists a dynamical system (X,F ,m, τ) and anf ∈ L1(X,m) such that

‖ supN∈S| 1N

N∑k=1

f τnk |‖1,∞ > C‖f‖1.

By the Conze and Banach principles (see Section 1.5), this implies

Corollary 1.5. The sequence of dth powers (d > 1) and the sequence of primes are persis-tently universally L1-bad.


1.4 Sequences of Convolution Operators

The third main result treats essentially the same problem in two different contexts. If wehave a family of operators Tn on a Banach space V such that Tnv converges in some weaksense for every v ∈ V , we might ask whether there exists a subsequence Tnk such that Tnkvconverges in some stronger sense for each v ∈ V . We begin by considering a recent result ofRosenblatt [24] on approximate identities:

Definition A sequence of functions φn ∈ L1(R) is an approximate identity on R if ‖φn ∗f −f‖1 → 0 as n→∞, for all f ∈ L1(R).

Theorem 1.6. ([24], Theorem 2.7) Let φn be an approximate identity on R consisting ofnonnegative functions. Then there is a sequence nk such that φnk ∗ f → f a.e. for everyf ∈ Lp(R), for all p > 1.

This result is the analogue of a theorem for weighted averages in an earlier paper by Bellow,Jones and Rosenblatt [5]. In that context, the most natural analogue to an approximateidentity is a sequence of probability measures µn such that for any ergodic dynamicalsystem (X,F ,m, T ) and any f ∈ L1, the weighted averages

µnf(x) :=∑j∈Z

f(T jx)µn(j) (1.4)

converge in the L1 norm to∫Xfdm. This is equivalent ([5], Proposition 1.7b and Corollary

1.8) to the Fourier condition µn(γ) → 0 ∀γ ∈ T \ 1. However, we (and they) require astronger condition:


Definition We say that a sequence µn of probability measures on Z has asymptoticallytrivial transforms if µn converges to 0 uniformly on every compact subset of T \ 12, orequivalently, if supγ∈T |(1− γ)µn(γ)| → 0.

Bellow, Jones and Rosenblatt [5] proved the following:

Theorem 1.7. Suppose µn is a sequence of probability measures on Z with asymptoticallytrivial transforms. Then there exists a subsequence nk such that µnkf(x) converges a.e.for every dynamical system (X,F ,m, T ) and every f ∈ Lp, p > 1.

We will prove weak-type (1,1) inequalities on R and on Z which extend Theorem 1.6 andTheorem 1.7 to L1:

Corollary 1.8. Let φn an approximate identity on R consisting of nonnegative functions.Then there is a sequence nk such that φnk ∗ f → f a.e. for every f ∈ L1(R).

Corollary 1.9. Suppose µn has asymptotically trivial transforms. Then there exists asubsequence nk such that µnkf(x) converges a.e. for every dynamical system (X,F ,m, τ)and every f ∈ L1(X).


1.5 The Banach Principle and the Conze Transference

Principle

With our major results stated, we turn to the background of this research. The topics ofthis thesis are the pointwise convergence or divergence for L1 functions of ergodic averageswhich are known to converge pointwise for all L2 functions3. The key to these questions, then,rests with the Banach principle, which for our purposes can be stated thus (see Theorem 5.1of [25] and Theorem 1 of [26]):

Theorem 1.10. Let (X,m) a probability space, 1 ≤ p ≤ 2, and Tn : Lp(X,m)→ Lp(X,m)a sequence of linear Lp contractions. If T ∗f(x) := supn≥1 |Tnf(x)| <∞ a.e. x ∈ X for everyf ∈ Lp(X), then f ∈ Lp(X) : limn→∞ Tnf exists a.e. is closed in the Lp(X) norm, andthere exists a constant C <∞ such that

m(x ∈ X : T ∗f(x) > λ) ≤ C

λp

∫X

|f |p dm ∀f ∈ Lp(X). (1.5)

2Here and in Chapter 2, we identify T with the unit circle rather than with R/Z, because working withpolynomials in γ will be made easier thereby.

3To be precise, the results of Section 1.4 involve passing to a subsequence of the averages, but the idea isthe same.


On the other hand, if for a sequence Tn of Lp contractions there does not exist C < ∞such that (1.5) holds, then there exists an f ∈ Lp(X) such that Tnf(x) diverges on a set ofpositive measure in X.

At first it may appear that this constant might depend on the particular dynamical sys-tem. However, it turns out that this is not the case for ergodic averages: we can transferweak maximal inequalities (and indeed, many other types of inequalities) from any ergodicdynamical system to the integers with the shift operator, and from the integers to any dy-namical system. This is known as the Conze principle [17].

Given a dynamical system (X,m, T ) and aN,jN∈N,j∈Z with∑

j∈Z |aN,j| < ∞ ∀N , we can

define the weighted ergodic averages on L1(X,m),

ANf(x) =∑j∈Z

aN,jf(T jx), (1.6)

and the corresponding convolution operators on `∞(Z),

ANϕ(k) =∑j∈Z

aN,jϕ(k − j) = ϕ ∗ µN(k)

where

µN :=∑j∈Z

aN,jδj. (1.7)

We first prove the Calderon transference principle, which takes us from the integers to anarbitrary dynamical system; this proof follows Bourgain [7].

Proposition 1.11. Let (X,m, T ) be a dynamical system; let aN,jN∈N,j∈Z with∑

j∈Z |aN,j| <∞ ∀N , and C0 <∞. Then for any 1 ≤ p ≤ ∞:

1. If ‖ supN |ψ ∗ µN |‖p ≤ C0‖ψ‖p ∀ψ ∈ `p, then ‖ supN |ANf |‖p ≤ C0‖f‖p ∀f ∈ Lp(X).

2. If ‖ supN |ψ ∗ µN |‖p,∞ ≤ C0‖ψ‖p ∀ψ ∈ `p, then ‖ supN |ANf |‖p,∞ ≤ C0‖f‖p ∀f ∈Lp(X).

Proof. We first consider the strong type maximal inequality. It is enough to show

‖ sup1≤N≤J

|ANf |‖p ≤ C0‖f‖p ∀f ∈ Lp(X),

independent of J ∈ N. We may further assume that the support of each µN is finite (theerror thus introduced can be made smaller than ε2−N), and choose M ∈ N so that suppµN ⊂ [−M,M ] for all N ≤ J . Fix x ∈ X and a large K ∈ N, and define ϕ on Z by

ϕ(n) =

f(T−nx) if |n| ≤ K +M,0 otherwise.


Then Ajf(T kx) = ϕ ∗ µN(−k) for all |k| ≤ K and all N ≤ J . This completes the proof forp =∞; for p <∞,∑

|k|≤K

sup1≤N≤J

|ANf(T kx)|p =∑|k|≤K

sup1≤N≤J

|ϕ ∗ µN(−k)|p ≤ ‖ sup1≤N≤J

|ϕ ∗ µN |‖pp

≤ Cp0‖ϕ‖pp

= Cp0

∑|n|≤K+M

|f(T−nx)|p.

Integrating over x ∈ X,

‖ sup1≤N≤J

|ANf |‖pp ≤ Cp0

2K + 2M + 1

2K + 1‖f‖pp;

letting K →∞, we obtain

‖ sup1≤j≤J

|Ajf |‖p ≤ C0‖f‖p.

For the weak type inequality, we similarly derive

λp||k| ≤ K : sup1≤N≤J

|ANf(T kx)| > λ| ≤ Cp0‖ϕ‖pp

and integrate this in the same manner.

The transfer in the other direction requires a fact about ergodic dynamical systems, theRohlin Tower Lemma. We use the version in [2] (Lemma 4.1):

Lemma 1.12. Let T : X → X be an ergodic, invertible measure-preserving transformationon a non-atomic measure space (X,F ,m), N a positive integer, and ε > 0. Then there existsA ∈ F such that A, TA, T 2A, . . . , TN−1A are pairwise disjoint and cover X up to a set ofmeasure less than ε.

With this in hand, we may treat the sets T kA as the discrete interval 0, . . . , N − 1,and thus the transfer of inequalities from an ergodic dynamical system to Z is trivial.

Given these transfer principles, the extensions of ergodic theorems considered in this the-sis can be reduced to establishing the validity of the weak (1,1) maximal functions for therelevant sequences of convolution operators on `1(Z).


1.6 Two Recurring Tools

An important tool in our positive results is the discrete version of the classical Calderon-Zygmund decomposition, by which we can treat a function ϕ ∈ `1(Z) as the sum of piecessupported on dyadic blocks, with a uniform size for their averages over such blocks, plus an`∞ error term. That is:

Theorem 1.13. Let ϕ ∈ `1(Z), and λ > 0. Then ϕ = g + b, where

• ‖g‖∞ ≤ λ

• b =∑

(s,k)∈B

bs,k for some index set B ⊂ N0 × Z

• bs,k is supported on the discrete dyadic cube Qs,k = [k2s, (k + 1)2s) ∩ Z

• Qs,k : (s, k) ∈ B is a disjoint collection

• ‖bs,k‖1 ≤ λ|Qs,k| = λ2s for all (s, k) ∈ B

•∑

x bs,k(x) = 0 for all (s, k) ∈ B

•∑

(s,k)∈B

|Qs,k| ≤C

λ‖ϕ‖1 (C a universal constant).

Proof. (This proof is adapted from lecture notes by M. Christ.) Consider the collection of alldiscrete dyadic cubes Q such that

∑x∈Q |ϕ(x)| > λ|Q|; since ‖ϕ‖1 <∞, clearly this is a finite

collection. Thus we may order these cubes by inclusion and take a maximal subcollectionindexed by B ⊂ N× Z. Now we define

g(x) :=

ϕ(x), x /∈

⋃B

Qs,k

2−s∑x∈Qs,k

ϕ(x), x ∈ Qs,k, (s, k) ∈ B;

bs,k(x) := ϕ(x)− g(x), x ∈ Qs,k.

By the maximality of dyadic cubes in this collection, clearly∑x∈Qs,k

|ϕ(x)| ≤ 2λ2s for each

(s, k) ∈ B, and therefore ‖g‖∞ ≤ 2λ. From these it follows that∑

x bs,k(x) = 0 and‖bs,k‖1 ≤ 4λ2s for all (s, k) ∈ B. As we may replace λ with λ′ = λ/4 at the start, it onlyremains to show the last property. Note that by the selection of the Qs,k,∑

(s,k)∈B

|Qs,k| <∑

(s,k)∈B

λ−1∑x∈Qs,k

|ϕ(x)| ≤ λ−1‖ϕ‖1

since the collection is disjoint by maximality.


Another tool which arises in more than one of these papers is Chernoff’s Inequality,a large-deviations inequality that is quite simple to work with. Its proof, based on theexponential moment method, can be found in [29] (Theorem 1.8).

Theorem 1.14. Let XnNn=1 be jointly independent random variables with |Xn| ≤ 1 and

EXn = 0. Let X =N∑n=1

Xn, and σ2 = VarX = EX2. Then for any λ > 0,

P(|X| ≥ λσ) ≤ 2 max(e−λ2/4, e−λσ/2).

1.7 L2 Theory

Following Rosenblatt and Wierdl’s exposition [25], we note that the question of meanconvergence of ANf as defined in (1.6) reduces to the pointwise convergence of the Fouriertransform of the corresponding `1 function µN from (1.7). That is:

Proposition 1.15. The weighted ergodic averages ANf converge in the L2 norm, for everydynamical system (X,F ,m, T ) and every f ∈ L2(X), if and only if the exponential sums

µN(θ) =∑j∈Z

aN,je2πijθ

converge for every θ ∈ T.

Proof. For any (X,F ,m, T ) and any fixed f ∈ L2(X), the sequence 〈T nf, f〉 is a positivedefinite sequence in the notation of the Herglotz theorem ([21], Section 7.6), and thereforethere exists a positive measure ν on T such that 〈T nf, f〉 =

∫T e

inθ dν(θ) for all n ∈ Z;furthermore, ν(T) = ‖f‖2

2 <∞. Therefore

‖ANf − AMf‖2L2(X) =

∑j,k

(aN,j − aM,j)(aN,k − aM,k)〈T jf, T kf〉

=

∫T

∑j,k

(aN,j − aM,j)(aN,k − aM,k)e2πi(j−k)θ dν(θ)

=

∫T

∣∣∣∣∣∑j

(aN,j − aM,j)e2πijθ

∣∣∣∣∣2

dν(θ)

=

∫T|µN(θ)− µM(θ)|2 dν(θ).

If µN(θ) converges for every θ as N → ∞, then by the Bounded Convergence Theorem wesee that ANf is a Cauchy sequence in L2(X).


For the other direction, consider the dynamical system X = [0, 1) with Lebesgue mea-sure and Tx = x + θ0 mod 1, and the function f(x) = e2πix. Then ANf = µN(θ0)f , so ifµN(θ0) diverges, then ANf does as well.

The pointwise theory is more delicate, and makes use of the transfer principle for varioustypes of inequalities (as in Section 1.5) to reduce to questions of exponential sums. Thesimplest of the cases we are concerned with is that of the Bernoulli random averages (1.3).We note that the expected value

Eω(AωNf(x)) = Nα−1

N∑n=1

n−αf(T nx)

is a type of Cesaro mean for the standard ergodic averages, and thus its pointwise convergenceis assured for any integrable f . Thus we need only concern ourself with the mean 0 differencesηn(ω) := ξn(ω)− n−α and the associated exponential sums

νωN(θ) := Nα−1

N∑n=1

ηn(ω)e2πinθ.

Applying Chernoff’s Inequality at a suitable number of points in T quickly establishes thatfor any ε > 0, with probability 1 we have

‖νωN‖∞ ≤ CωN(α−1)/2+ε ∀N ∈ N.

This more than suffices to show that for each ω in this set of probability 1, for each dynamicalsystem, each f ∈ L2(X) and each ρ > 1, the averages Aωbρkcf(x)k∈N must converge for a.e.x, which implies the pointwise convergence of AωNf .

The cases of the averages along polynomials and along the primes proceed by a simi-lar method, although complicated by the fact that the associated exponential sums do notconverge to 0 at rational points. The analysis of these exponential sums makes use of theHardy-Littlewood circle method (see e.g. [31]), in which one type of estimate is obtainednear rational numbers with small denominators (the major arcs) and another is obtained onthe rest of the circle (the minor arcs).

It is relatively straightforward to pass from the exponential sums corresponding to thesearithmetic sequences to certain simple functions on T supported on the major arcs. Themost substantial part of Bourgain’s argument in [7]-[9] consists of proving maximal and os-cillational inequalities for the weighted averages whose Fourier transforms are those simplefunctions.


Bellow, Jones and Rosenblatt [5] developed the L2 theory for the averages (1.4) by con-sidering the square functions

Sf(x) :=

(∞∑n=1

|µnf(x)− σnf(x)|2)1/2

,

where σnf is a sequence of standard ergodic averages. The inequality ‖Sf‖L2(X) . ‖f‖L2(X)

then guarantees the a.e. convergence of the sequence µnf , and is implied by the boundednessof ∥∥∥∥∥

∞∑n=1

|µn − σn|2∥∥∥∥∥L∞(T)

.

Since we may assume µn(0) = σn(0) = 1, and since we have assumed the µn have asymptoti-cally trivial transforms, we can indeed find a subsequence nk along which the above quantityis finite.

1.8 L1 Theory: Heuristics

As noted in Section 1.5, the ergodic theorems investigated in this thesis are equivalent tothe validity of weak (1,1) maximal inequalities for certain sequences of convolution operatorson `1(Z). That is, given a sequence µn ∈ `1, we are trying to decide whether there existsC <∞ such that

|k ∈ Z : supn|ϕ ∗ µn(k)| > λ| ≤ C

λ‖ϕ‖1 ∀ϕ ∈ `1. (1.8)

The approach to this inequality for the standard ergodic averages (µn = 1n

∑nj=1 δj) is the

maximal ergodic theorem. The classical proofs of the maximal ergodic theorem make useof identities like (N + 1)AN+1f = f T + NANf T , and thus can only be extended tocases like averages over block sequences (as in [3]). One can also use a weak (1,1) maximalinequality for averages along a sequence to obtain one for averages along a subsequence ofpositive density. As noted in Section 1.2, these were the only types of L1 results publishedprior to 2007.

In order to obtain the results of Sections 1.2 and 1.4, we pursue a quite different method ofobtaining (1.8). In each case, a key part of the left-hand side will have an exceptionally smallL2 norm after the function ϕ is decomposed into certain pieces. The “gain” from this needsto overcome a factor of at least |supp µn|1/2 for each n, as we would expect when trying touse an L2 bound for L1 purposes.


This technique was first developed in the context of singular integral theory, in papersby Fefferman [18] on the Lp convergence of partial sums of Fourier series in n dimensions,and further developed by Christ [14] to prove weak (1,1) results for a number of maximaloperators which were not amenable to earlier techniques. Another application to sphericalmeans may be found in [27]. It was extended to the discrete setting, with a few additionalcomplications, in the aforementioned paper of Urban and Zienkiewicz [30].

The exposition developed in a recent note of Christ [13] is the cleanest version of this ap-proach, as it does not require the full Calderon-Zygmund decomposition, only a simple de-composition by height. We will reproduce (a toy version of) the argument here for heuristicpurposes, noting that the last hypothesis does not apply in the cases under consideration.

Theorem 1.16. Let G be a discrete abelian group. Let µk, νk : G→ C satisfy

The maximal operator supk|f | ∗ |νk| is of weak type (1,1) on G, (1.9)

each µk satisfies

|supp (µk)| ≤ C2k, (1.10)

and

‖µk − νk‖∞ ≤ C2−k/2. (1.11)

Then the maximal operator supk∈N |f ∗ µk| is of weak type (1, 1) on G.

Proof. Decompose f =∑

j∈Z fj where fj(x) :=

f(x), 2j ≤ |f(x)| < 2j+1,0 otherwise.

By scaling f , we may assume without loss of generality that λ = 1 in (1.8). Considerthe exceptional sets

Ej :=

j⋃k=1

(supp fj) + (supp µk), E :=⋃j∈N

Ej.

Now

|Ej| ≤j∑

k=1

C2k|supp fj| = C2j|supp fj|, |E| ≤ C∑j

2j|supp fj| ≤ C‖f‖1,

so this is a small enough exceptional set. For x /∈ E,

|f ∗ µk(x)| ≤∑j<k

|fj ∗ µk(x)|

≤∑j<k

|fj| ∗ |νk(x)|+∑j<k

|fj ∗ (µk − νk)(x)|

≤ |f | ∗ |νk(x)|+∑j<k

|fj ∗ (µk − νk)(x)|.


The weak maximal inequality corresponding to the first term holds by assumption. For thesecond term we introduce the square functions

Gs(x) :=

(∑k

|fk−s ∗ (µk − νk)(x)|2)1/2

and bound

supk

∑j<k

|fj ∗ (µk − νk)(x)| ≤∞∑s=1

Gs(x).

Then

‖Gs‖22 =

∑k

‖fk−s ∗ (µk − νk)‖22 ≤

∑k

‖µk − νk‖2∞‖fk−s‖2

2

≤∑k

C2−k‖fk−s‖∞‖fk−s‖1 ≤ C∑k

2−s‖fk−s‖1 ≤ C2−s‖f‖1,

so that ‖∞∑s=1

Gs‖22 ≤ C‖f‖1, which is the required bound.

Remark By Parseval’s identity, we can see that the Fourier condition involved (‖µk−νk‖∞ .|supp µk|−1/2) is optimal for probability measures, and indeed it is rare to find sequences ofmeasures for which this applies. (In [13], the measures are obtained from special measureson finite fields via Freiman isomorphisms.) The positive results in this thesis will thereforefind different methods of exploiting the L2 theory to obtain a strong enough bound.

The heuristics for the result of Section 1.3 are far more subtle, and will be discussed later inSection 4.2.

16

Chapter 2

Sequences of Convolution Operators

2.1 Approximate Identities

We shall begin with the results of Section 1.4, since they use a simpler version of theargument that is used to prove the results in Section 1.2– namely, because we are able topass to subsequences, we can obtain sufficiently strong Fourier estimates in this case.

Each L1 subsequence result makes use of the following additional technique: given theCalderon-Zygmund decomposition of a function f = g +

∑s bs, we classify s as “small”,

“large” or “intermediate” with respect to each term of our subsequence φn or µn. For s“large”, we can use a covering lemma to handle the terms; for s “small”, we will use cancel-lation properties of bs; and since for each s there will be only one n for which it counts as“intermediate”, we can handle these terms with a trivial estimate. This idea plays a role in[28] as well as other papers.

Let |E| denote the Lebesgue measure of a measurable set E ⊂ R.

Proposition 2.1. Let φn an approximate identity on R consisting of nonnegative functions.Then there is a sequence nk such that we have the weak-type maximal inequality

|x : supk|φnk ∗ f(x)| > λ| ≤ C

λ‖f‖L1(R) ∀f ∈ L1(R).

Given Theorem 1.6, this implies Corollary 1.8.

Proof. By Proposition 2.4 of [24], there exists another approximate identity ψn consisting ofnonnegative functions of L1 norm 1 whose supports are eventually contained in any neigh-borhood of 0, such that ‖φn−ψn‖1 → 0. We can furthermore require that each ψn be a stepfunction with steps [rni, rni+ 1], for some positive real numbers rn → 0.

CHAPTER 2. SEQUENCES OF CONVOLUTION OPERATORS 17

We may pass to a subsequence such that∑‖φnk − ψnk‖1 < ∞, in which case the weak-

type maximal inequality for supk |(φnk − ψnk) ∗ f | holds trivially, and it suffices to provea weak-type maximal inequality for supk |ψnk ∗ f |. Thus we may assume without loss ofgenerality that φn has these properties from the start; that is,

φn = r−1n

Rn−1∑i=−Rn

ci,nχ[rni,rn(i+1)], (2.1)

ci,n ≥ 0, (2.2)Rn−1∑i=−Rn

ci,n = 1, (2.3)

Rnrn → 0. (2.4)

Finally, we may assume (by passing to a subsequence) that rn+1Rn+1 ≤ 12R−1n rn, ∀n ≥ 1.

Given f ∈ L1 and λ > 0, we perform the standard Calderon-Zygmund decomposition on R:f = g+

∑(s,k)∈B bs,k, bs,k supported on dyadic cubes Qs,k in R, with the analogous properties

to Theorem 1.13 (except that s ∈ Z here).

Then as usual in Calderon-Zygmund arguments, we let Q?s,k denote the interval with the

same center as Qs,k but length 3 · 2s. Noting that |φn ∗ g(x)| ≤ λ for all n and x,

|supn|φn ∗ f(x)| > 2λ| ≤ |sup

n|φn ∗ g(x)| > λ|+ |sup

n|φn ∗ b(x)| > λ|

≤ 0 +∑s,k

|Q?s,k|+

1

λ

∑n

‖φn ∗ b‖L1(R\∪Q?s,k)

≤ C

λ‖f‖1 +

1

λ

∑(s,k)∈B

∑n

‖φn ∗ bs,k‖L1(R\Q?s,k)

and we claim there is an absolute constant C ′ such that for all s and j,∑n

‖φn ∗ bs,k‖L1(R\Q?s,k) ≤ C ′‖bs,k‖1. (2.5)

(This clearly suffices to obtain the weak type (1,1) bound.)

Let n(s) := maxn : Rnrn ≥ 2−s. If n > n(s), then x : |φn ∗ bs,k(x)| > 0 ⊂ Q?s,k.

If n ≤ n(s), then we control the support of φn ∗ bs,k by noting that φn is constant on everyinterval (rni, rn(i+ 1)), so that x−Qs,k ⊂ [rni, rn(i+ 1)] =⇒ φn ∗ bs,k(x) = 0. Therefore

|x : |φn ∗ bs,k(x)| > 0| ≤

∣∣∣∣∣i=Rn⋃i=−Rn

(rni− 2−s, rni+ 2−s)

∣∣∣∣∣ ≤ 4Rn2−s.


Now for any x ∈ R,

|φn ∗ bs,k(x)| ≤ r−1n

∫ ∞−∞|bs,k(x+ y)|dy = r−1

n ‖bs,k‖1,

and thus ∑n

‖φn ∗ bs,k‖L1(R\Q?s,k) ≤ ‖φn(s) ∗ bs,k‖1 +∑n<n(s)

‖φn ∗ bs,k‖1

≤ ‖bs,k‖1 +∑n<n(s)

4Rnr−1n 2−s‖bs,k‖1

≤ ‖bs,k‖1 +∑n<n(s)

8(Rn+1rn+1)−12−s‖bs,k‖1

≤(1 + 16(Rn(s)rn(s))

−12−s)‖bs,k‖1 ≤ 17‖bs,k‖1

where the third and fourth inequalities are due to the condition Rn+1rn+1 ≤ 12R−1n rn ≤

12Rnrn. This completes the proof.

Remark This proof, as well as the argument in [24], generalizes straightforwardly to Rd.

2.2 Weighted Averages

We now turn to the result on weighted ergodic averages. Let µn a sequence of prob-ability measures on Z, and recall (from Section 1.4) that we say µn has asymptoticallytrivial transforms if supγ∈T |(1− γ)µn(γ)| → 0.

Theorem 2.2. Suppose µn has asymptotically trivial transforms. Then there is a subse-quence nk which obeys the weak type maximal inequality

|x : supk|ϕ ∗ µnk(x)| > λ| ≤ Cλ−1‖ϕ‖`1(Z) ∀ϕ ∈ `1(Z).

Given Theorem 1.7, this implies Corollary 1.9).

Proof. We may write µn = µ′n + ηn, where µ′n is compactly supported and∑∞

n=1 ‖ηn‖`1(Z) <∞. Then

|x : supn|ϕ ∗ ηn(x)| > λ| ≤ λ−1

∞∑n=1

‖ϕ ∗ ηn‖`1(Z)

≤ λ−1

(∞∑n=1

‖ηn‖`1(Z)

)‖ϕ‖`1(Z).


Now |µ′n(γ)| ≤ |µn(γ)|+2‖ηn‖1, and so µ′n converges to 0 uniformly on every compact subsetof T \ 1. Thus we may assume that µn is compactly supported for each n.

Furthermore, if the union of these supports were compact, then it is easy to see (by Parseval’sTheorem) that ‖µn‖`1(Z) → 0 and we may choose a subsequence such that

∑k ‖µnk‖`1(Z) <

∞; such a subsequence would trivially satisfy a weak maximal inequality.

We may therefore assume that the union of the supports of the µn is unbounded, and setS(n) := mins ≥ 0 : supp µm ⊂ [−2s, 2s] ∀m ≤ n, and N(s) := minn : S(n) > s. Notethat S(N(s)− 1) ≤ s < S(N(s)) for all s.

Since we will want the cancellation properties of µn+1 to overcome the size of the supportof µn, we now choose an increasing subsequence nk such that supγ∈T |(1 − γ)µnk(γ)| ≤2−2S(nk−1)−2k and such that S(n) is strictly increasing. For ease of notation, we will considerµn to be this subsequence (and redefine S(n) accordingly). Thus we have the properties

supp µn ⊂ [−2S(n), 2S(n)] (2.6)

supγ∈T|(1− γ)µn(γ)| ≤ 2−2S(n−1)−2n. (2.7)

Given ϕ ∈ `1 and λ > 0, we perform the discrete Calderon-Zygmund decomposition (Theo-rem 1.13): ϕ = g+

∑(s,k)∈B bs,k. Let bs =

∑k bs,k for each s, and let Q?

s,k denote the intervalwith the same center as Qs,k and 3 times the length.

Since ‖µn ∗ g‖∞ ≤ ‖µn‖1‖g‖∞ ≤ λ for all n,

|x : supn|µn ∗ ϕ(x)| > 3λ| ≤ |x : sup

n|µn ∗ g(x)| > λ|+ |x : sup

n|µn ∗ b(x)| > 2λ|

≤ 0 +∑s,k

|Q?s,k|+ |x 6∈

⋃s,k

Q?s,k : sup

n|µn ∗ b(x)| > 2λ|

≤ C

λ‖ϕ‖1 + |x : sup

n|µn ∗

∑s<S(n)

bs(x)| > 2λ|,

because s ≥ S(n) =⇒ supp µn ∗ bs,k ⊂ Q?s,k.

We will be able to use (2.7) to our advantage when considering µn ∗ bs for s < S(n − 1),since in this case the size of the supports will be controlled relative to the relevant Fouriermultiplier norm in `2. This leaves the intermediate terms S(n − 1) ≤ s < S(n); but asmentioned in the introduction, these can be controlled by a simple `1 bound.


(This is a standard Calderon-Zygmund argument so far.) Now for the other sum, we canwrite

1− σn(γ) = (1− γ)2S(n−1)+n−1∑

i=0

(1− i2−S(n−1)−n)γi

in order to see that

‖µn(1− σn)‖∞ ≤ 2S(n−1)+n supγ|(1− γ)µn(γ)| ≤ 2−S(n−1)−n.

Here, using a version of the technique from Section 1.8, we will use our extremely strong `2

estimate (gained by comparing the Fourier bound for µn to bs with s < S(n− 1)) to obtaina weak `1 estimate:

|x : supn|(µn − µn ∗ σn) ∗

∑s<S(n−1)

bs(x)| > λ| ≤ λ−2

∥∥∥∥∥∥supn

∣∣∣∣∣∣(µn − µn ∗ σn) ∗∑

s<S(n−1)

bs

∣∣∣∣∣∣∥∥∥∥∥∥

2

2

≤ λ−2∑n

∥∥∥∥∥∥(µn − µn ∗ σn) ∗∑

s<S(n−1)

bs

∥∥∥∥∥∥2

2

= λ−2∑n

∥∥∥∥∥∥µn(1− σn)∑

s<S(n−1)

bs

∥∥∥∥∥∥2

2

≤ λ−2∑n

‖µn(1− σn)‖2∞

∥∥∥∥∥∥∑

s<S(n−1)

bs

∥∥∥∥∥∥2

2

≤ λ−2∑n

2−2S(n−1)−2n

∥∥∥∥∥∥∑

s<S(n−1)

bs

∥∥∥∥∥∥2

2

= λ−2∑n

2−2S(n−1)−2n∑

s<S(n−1)

‖bs‖22

by disjointness of supports. Now since ‖bs‖∞ ≤ λ2s,

≤ λ−2∑n

2−2S(n−1)−2n∑

s<S(n−1)

λ2s‖bs‖1

≤ λ−1∑n

2−S(n−1)−2n‖b‖1 ≤C

λ‖ϕ‖1.

This completes the proof.


Remark This proof generalizes straightforwardly to measure-preserving Zd-actions, andindeed, to actions by finitely generated abelian groups (this requires defining the Calderon-Zygmund decomposition on such a group, using the dyadic cubes from [15]). Note thatTheorem 1.7 generalizes to this case for p = 2, which is enough to establish it for 1 ≤ p ≤ 2;see Theorem 2.4 in [5].

2.3 Non-Uniform Fourier Decay

We can see that µn(γ)→ 0 ∀γ 6= 1 is too weak a condition for measures in general, if wealso lift the positivity assumption1. Indeed, later in this thesis I prove that the averages alongthe squares µn = 1

n

∑nk=1 δk2 are persistently universally L1-bad; while µn does not converge

to 0 everywhere, the very regular behavior of these exponential sums can be used to gener-ate another sequence of measures whose exponential sums do converge to 0 everywhere on T.

Consider the sequence µn(x) := µn(x)e2πin−1/2x. It can be verified by the Hardy-Littlewoodcircle method [31] that νk(γ) → 0 ∀γ ∈ T, since shifting the Fourier transform by n−1/2

prevents every rational number from entering its own major arc, and since there are onlyfinitely many peaks of a given height to pass over a given point. Because µn ∗ f(x) =

e2πin−1/2x(µn ∗ (e−2πin−1/2·f)(x)), a weak maximal inequality that fails for µnk will also failfor µnk .

However, it is (to the best of the author’s knowledge) still open whether there exists apersistently universally L1-bad sequence of probability measures with Fourier transformsµn(γ) converging pointwise to 0 for all γ 6= 1.

1This is not an entirely ridiculous thing to do: the L2 version of Theorem 1.7 does not in fact require the µnto be positive measures. Any sequence of uniformly `1-bounded measures on Z with supγ∈T |(1−γ)µn(γ)| → 0will suffice to prove an analogue of the crucial Lemma 1.12 from [5].

23

Chapter 3

Random Subsequences of Density 0

3.1 Reductions

We now turn to Theorem 1.3. After reducing the problem to that of a weak maximalinequality on Z, we will use another variant of the techniques from Section 1.8 to prove thisinequality under an assumption about µ

(ω)n ; in Section 3.3, we use Chernoff’s Inequality to

establish that with probability 1, these random measures do indeed satisfy that assumption.

Recall our setup: let Ω be a probability space, and define independent Boolean randomvariables ξn(ω) : n ∈ N on Ω with P(ξn = 1) = n−α and P(ξn = 0) = 1 − n−α. We beginby noting that the statement that n : ξn(ω) = 1 is a universally L1-good sequence withprobability 1 is equivalent to the existence of Ω′ ⊂ Ω with probability 1 such that for eachω ∈ Ω′, we have an L1 pointwise ergodic theorem for the averages

A(ω)N f(x) = Nα−1

N∑n=1

ξn(ω)f(T nx).

For this, it suffices to point out that with probability 1, Nα−1∑N

n=1 ξn(ω)→ C ∈ (0,∞), ascan be easily derived from Chernoff’s Inequality or Kolmogorov’s version of the Strong Lawof Large Numbers.

As noted in Section 1.7, there is a set Ω1 ⊂ Ω with P(Ω1) = 1 such that for ω ∈ Ω1 we

have a.e. convergence of A(ω)N f for all f ∈ L2(X), which is dense in L1(X). Theorem 1.3

thus reduces to proving on a set of probability 1 the weak maximal inequality

‖ supN|A(ω)

N f |‖1,∞ ≤ Cω‖f‖1 ∀f ∈ L1(X). (3.1)

CHAPTER 3. RANDOM SUBSEQUENCES OF DENSITY 0 24

As usual, it is enough to take this supremum over the dyadic subsequence 2j : j ∈ N, since

0 ≤ A(ω)N f ≤ 2A

(ω)

2j+1f for f ≥ 0 and 2j ≤ N < 2j+1. Now if we define the `1(Z) functions

µ(ω)j (n) :=

2(α−1)jξn(ω), 1 ≤ n ≤ 2j

0 otherwise

Eµj(n) :=

2(α−1)jEξn, 1 ≤ n ≤ 2j

0 otherwise

ν(ω)j (n) := µ

(ω)j (n)− Eµ(ω)

j (n),

by the Calderon transference principle (Proposition 1.11), it suffices to prove that withprobability 1 in Ω,

‖ supj|ϕ ∗ µ(ω)

j |‖1,∞ ≤ Cω‖ϕ‖1 ∀ϕ ∈ `1(Z). (3.2)

We will use µ to denote the reflection of a function µ about the origin; because

〈f ∗ µ, g ∗ µ〉 = 〈f ∗ µ ∗ µ, g〉,

this will be an important object. (It would be standard to use the notation µ∗, but thisbecomes unwieldy when using other superscripts as above.)

As noted before, the Fourier bounds on ν(ω)j are not good enough to use Theorem 1.16.

Instead, following the approach of Urban and Zienkiewicz [30], we will achieve our `2 bound

by first decomposing µ(ω)j ∗ µ

(ω)j into a point mass at 0 plus a very small `∞ error1. As most

of the `2 mass will be concentrated in the point mass, this will afford us a better bound thanwe would achieve by the Fourier transform.

3.2 Calderon-Zygmund Argument

We use a generalization of a deterministic argument from [30] (compare the hypothesesof Theorem 1.16):

Proposition 3.1. Let µj and νj be sequences of functions in `1(Z), with ‖µj‖1 ≤ 1. Letrj := |supp µj| and suppose supp µj, supp νj ⊂ [−Rj, Rj]. Assume there exist ε > 0 andA,A0, A1 <∞ such that

∑j≤k rj ≤ Ark ∀k ∈ N and

|νj ∗ νj(x)| ≤ A0r−1j δ0(x) + A1R

−(1+ε)j , ∀x ∈ Z. (3.3)

If there exists C <∞ such that for all ϕ ∈ `1(Z),

‖ supjϕ ∗ |µj − νj|‖1,∞ ≤ C‖ϕ‖1, (3.4)

1In [30], there is an additional term as well– a slowly varying function of intermediate size.


then there exists C ′ <∞ such that

‖ supj|ϕ ∗ µj|‖1,∞ ≤ C ′‖ϕ‖1 ∀ϕ ∈ `1(Z). (3.5)

Proof. We will follow the argument in Section 3 of [30], which makes use of a Calderon-Zygmund type decomposition of ϕ depending on the index j. We begin with the standard

decomposition (Theorem 1.13) at height λ > 0: ϕ = g+b = g+∑

(s,k)∈B bs,k. Let bs =∑k

bs,k.

We will divide∑s

bs into two parts, splitting at the index

s(j) := mins : 2s ≥ Rj.We begin by noting that since ‖g ∗ µj‖∞ ≤ λ,

|x : supj|ϕ ∗ µj(x)| > 3λ| ≤ |sup

j|g ∗ µj| > λ|+ |sup

j|b ∗ µj| > 2λ|

≤ 0 + |supj|b ∗ (µj − νj)| > λ|+ |sup

j|∞∑s=0

b ∗ νj| > λ|.

Now |b ∗ (µj − νj)(x)| ≤ |b| ∗ |µj − νj|(x), so by (3.4),

|supj|b ∗ (µj − νj)| > λ| ≤ C

λ‖b‖1 ≤

C

λ‖ϕ‖1.

As usual in Calderon-Zygmund arguments, note that for s > s(j),

supp (bs,k ∗ νj) ⊂ Qs,k + [−Rj, Rj] ⊂ Q∗s,k,

the interval with the same center as Qs,k and 3 times the length. Since∑(s,k)∈B

3|Qs,k| ≤C

λ‖ϕ‖1,

we have reduced the problem to obtaining a bound on the final term. We will first attemptthis directly for heuristic purposes, and then modify our setup for the actual argument. ByChebyshev’s Inequality,

|x : supj|s(j)∑s=0

bs ∗ νj(x)| > λ| ≤ λ−2∑x

supj|s(j)∑s=0

bs ∗ νj(x)|2

≤ λ−2∑j

∥∥∥∥∥∥s(j)∑s=0

bs ∗ νj

∥∥∥∥∥∥2

`2

= λ−2∑j

s(j)∑s,t=0

〈bs ∗ νj, bt ∗ νj〉`2

and we will use our estimate on the convolution product νj ∗ νj:


and B(j) by summing over one or both indices, respectively. Then

|supj|s(j)∑s=0

bs ∗ νj| > 3λ| ≤ |supj|s(j)∑s=0

b(j)s ∗ (νj − µj)| > λ|+ |sup

j|s(j)∑s=0

b(j)s ∗ µj| > λ|

+|supj|s(j)∑s=0

B(j)s ∗ νj| > λ|.

We control the first term using (3.4), since |b(j)| ≤ |b|; and for the second term,

|supj|s(j)∑s=0

b(j)s ∗ µj| > λ| ≤

∑j

|x : |b(j) ∗ µj(x)| > 0|

≤∑j

|supp µj| · |x : |b(x)| > λrj|

=∑j

rj∑k≥j

|x : λrk < |b(x)| ≤ λrk+1|

=∑k

|x : λrk < |b(x)| ≤ λrk+1|∑j≤k

rj

≤ A

λ

∑k

λrk|x : λrk < |b(x)| ≤ λrk+1|;

now since this sum is a lower sum for |b|, it is ≤ Aλ‖b‖1 ≤ C

λ‖ϕ‖1.

We proceed with the final term just as we tried before, since Lemma 3.2 applies to theB

(j)s as well as to the bs. We thus find

|supj|s(j)∑s=0

B(j)s ∗ νj| > λ| ≤ A0λ

−2∑j

r−1j ‖B(j)‖2

2 + 10A1λ−1∑j

log2(2Rj)R−εj ‖B(j)‖1

≤ A0λ−2∑x

∑j

r−1j |B(j)(x)|2 +

C

λ‖ϕ‖1.

Now∑j≤k

rj ≤ Ark ∀k ∈ N implies ∃N s.t. rj+n ≥ 2rj ∀j ∈ N, n ≥ N , which implies∑∞j=k r

−1j ≤ A′r−1

k ; thus for each x ∈ Z,∑j

r−1j |B(j)(x)|2 ≤

∑j:λrj≥|b(x)|

r−1j |b(x)|2 ≤ A′λ|b(x)|

so the sum is ≤ Cλ‖ϕ‖1 and the proof of Proposition 3.1 is complete.


3.3 Probabilistic Lemma

Having established Proposition 3.1, it remains to show that the random measures µ(ω)j

and ν(ω)j satisfy the assumptions with probability 1. Note first that Rj = 2j+1, and with

probability 1,

rj = 2(1−α)j‖µ(ω)j ‖1 =

∑1≤n≤2j

ξn(ω) 2(1−α)j.

It merely remains to establish the bound (3.3) on ν(ω)j ∗ ν

(ω)j , which we prove via the following

lemma:

Lemma 3.3. Let E ⊂ Z, and let Xnn∈E be independent random variables with |Xn| ≤1 and EXn = 0. Assume that

∑n∈E(VarXn)2 ≥ 1. Let X be the random `1 function∑

n∈E Xnδn. Then for any θ > 0,

P

(supk 6=0|X ∗ X(k)| ≥ θ(

N∑n=1

(VarXn)2)1/2

)≤ 4|E|2 max(e−θ

2/16, e−θ/4). (3.6)

Proof. Fix k 6= 0. Then

X ∗ X(k) =∑

n∈E∩E−k

XnXn+k =∑n∈E

Yn,

where EYn = 0 and |Yn| ≤ 1 (of course Yn ≡ 0 if n + k /∈ E). We want to apply Chernoff’sInequality (Theorem 1.14), but the Yn are not independent.

However, we can easily partition E into two subsets E1 and E2 such that Ei ∩ (Ei − k) = ∅for each i; then within each Ei, the Yn depend on distinct independent random variables, sothey are independent.

Now∑n∈Ei

Yn has variance σ2i =

∑n∈Ei

VarXnVarXn+k ≤∑n∈E

(VarXn)2 by Holder’s Inequality.

Chernoff’s Inequality states that for any λ > 0,

P(|∑n∈Ei

Yn| ≥ λσi) ≤ 2 max(e−λ2/4, e−λσi/2).

Take λi = θσ−1i (∑

n∈E(VarXn)2)1/2; then λiσi = θ(∑

n∈E(VarXn)2)1/2 ≥ θ and λi ≥ θ, so

P(|X ∗ X(k)| ≥ 2θ(∑n∈E

(VarXn)2)1/2) ≤2∑i=1

P(|∑n∈Ei

Yn| ≥ λiσi)

≤ 4 max(e−θ2/4, e−θ/2).

Since this holds for each k 6= 0 and |supp X ∗ X| ≤ |E|2, the conclusion follows (replacing2θ with θ).


Corollary 3.4. Let ν(ω)j be the random measure defined as before, 0 < α < 1/2 and κ > 0.

Then there is a set Ω2 ⊂ Ω with P(Ω2 = 1) such that for each ω ∈ Ω2,

|ν(ω)j ∗ ν

(ω)j (x)| ≤ Cω2(α−1)jδ0(x) + Cω2(−3/2+α+κ)j. (3.7)

Proof. For the bound at 0, we use the fact that

ν(ω)j ∗ ν

(ω)j (0) = 22(α−1)j

2j∑n=1

(ξn(ω)− n−α)2

≤ 22(α−1)j

2j∑n=1

(ξn(ω) + n−α)

= 2 · 2(α−1)j + 22(α−1)j

2j∑n=1

(ξn(ω)− n−α)

so that

P(ν(ω)j ∗ ν

(ω)j (0) > 3 · 2(α−1)j) ≤ P

2j∑n=1

(ξn(ω)− n−α) > 2(1−α)j

≤ 2 exp(−2(1−α)j−1)

for j sufficiently large, by Chernoff’s inequality. The Borel-Cantelli Lemma implies that withprobability 1, ν

(ω)j ∗ ν(ω)

j (0) ≤ 3 · 2(α−1)j for j sufficiently large (depending on ω), so there

exists Cω with 0 ≤ ν(ω)j ∗ ν

(ω)j (0) ≤ Cω2(α−1)j for all j.

For the other term, we note that Var ξn ≤ n−α, so we set θ = 2κj and apply Lemma3.3:

P

22(1−α)j supk 6=0|ν(ω)j ∗ ν

(ω)j (k)| ≥ 2κj(

2j∑n=1

n−2α)1/2

≤ 4 · 22j exp(−2κj/4)

which sum over j. The Borel-Cantelli Lemma again proves the bound holds with probability1.

Note that for α < 1/2 and κ+ ε = 1/2− α,

2(− 32

+α+κ)j = R−(1+ε)j .

Therefore the measures ν(ω)j satisfy the bound (3.3) , for all ω ∈ Ω2. Since µ

(ω)j − ν

(ω)j = Eµj

is a weighted average of the regular ergodic averages, supj |ϕ∗Eµj| ≤ C supN |ϕ∗N−1χ[1, N ]|so that Birkhoff’s Ergodic Theorem implies the needed weak `1 bound; and the `∞ maximalinequality for µ

(ω)j is trivial. Thus Proposition 3.1 implies the weak maximal inequality (3.2),

and we have proved Theorem 1.3.


Remark This argument does not require the probabilities τn to obey a power law. If τn isdecreasing and if (

N∑n=1

τn

)−2√√√√ N∑

n=1

τ 2n ≤ CN−1−ε

for some ε > 0, C <∞ and all N , the sequence n : ξn(ω) = 1 will be universally L1-goodwith probability 1. Indeed, we need only assume that 0 ≤ ξn ≤ 1, Eξn is nonincreasing, andEξn . n−α for some α < 1/2.

Remark It is worthy of note that n : ξn = 1 indeed has Banach density 0 (with probability1) if the τn decrease more rapidly than some power law. Conveniently enough, a converseresult also holds:

Proposition 3.5. Let τn be a nonincreasing sequence of probabilities, and let ξn be inde-pendent Bernoulli random variables with P(ξn = 1) = τn. Then if τn = O(n−α) for someα > 0, the sequence of integers n : ξn = 1 has Banach density 0 with probability 1 in Ω;otherwise, it has Banach density 1 with probability 1 in Ω.

Proof. It is elementary to show that

2−rτmr(n+1) ≤ P

r(n+1)−1∑j=rn

ξj ≥ m

≤ 2rτmrn. (3.8)

(We majorize or minorize the ξj by i.i.d. Bernoulli variables and use the Binomial Theorem.)Then if τn = O(n−α), let K > 0 and fix m, r ∈ N such that mα > 1 and r > mK; the proba-bilities above are then summable, so the first Borel-Cantelli Lemma implies that on a set ΩK

of probability 1 in Ω, there exists an Mω such that for all n ≥ Mω,∑r(n+1)−1

j=rn ξj < m < rK

;

then it is clear that n : ξn = 1 has Banach density less than 3K−1. Let Ω′ =⋂K ΩK ; then

P(Ω′) = 1 and n : ξn = 1 has Banach density 0 on Ω′.

For the other implication, note that if τn 6= O(n−1/R), there exists a sequence nk with

nk+1 ≥ 2nk such that τnk ≥ n−1/Rk ; then

∞∑n=1

τRRn ≥ R−1

∞∑n=2

τRn ≥ R−1

∞∑k=2

(nk − nk−1)τRnk ≥ R−1

∞∑k=2

1

2=∞.

Thus the probabilities in (3.8) are not summable in n, for m = r = R. Since the variablesξn are independent, the second Borel-Cantelli Lemma implies that there is a set ΩR ofprobability 1 on which n : ξn(ω) = 1 contains infinitely many blocks of R consecutiveintegers. Therefore if τ(n) 6= O(n−α) for every α > 0, let Ω′ =

⋂R ΩR; on this set of

probability 1, n : ξn = 1 has Banach density 1.


Remark Because this argument makes no use of the Fourier transform, it can in fact beused to prove analogous results for more general group actions– in particular, we can provea random pointwise ergodic theorem for measure-preserving group actions of the form

A(ω)N f(x) := (

∑g∈SN

τg)−1∑g∈SN

ξgf(Tgx),

where G is a finitely generated virtually nilpotent group (i.e. a discrete group where iteratedsumsets of any finite set have polynomial growth eventually), Tg : X → X are measure-preserving transformations with Tg Th = Tgh for all g, h ∈ G, SN is a suitable Følnersequence in G, and τg are suitably decreasing probabilities. The analysis takes place on`1(G), and follows the same pattern as above.

A pointwise L2 ergodic theorem must first be proved for these random averages, since themethod of exponential sums cannot be applied to non-abelian groups. To this end, a com-binatorial argument suffices to bound the `1(G) norm of large convolution powers of therandom measure.

32

Chapter 4

Universally L1-Bad ArithmeticSequences

4.1 Introduction

Finally, we confront the results of Section 1.3; the methods of this section have an entirelydifferent flavor from those employed in the other parts of this thesis.

A few words on the structure of this exposition: In Section 4.2, we present a heuristicversion of the argument, in the case of the squares. Then in Section 4.3, we express thegeneral form of our result (Theorem 4.1) and prove that its conditions are indeed satisfiedby the dth powers and the sequence of primes. In Section 4.4, we present the main inductivestep (Proposition 4.3), show that it implies Theorem 4.1, and explain the structure of theinduction.

In Sections 4.5-4.7, we construct the various objects of the succeeding inductive step andprove several necessary lemmas about them. Section 4.8 brings these parts together andproves that the properties claimed in Proposition 4.3 do indeed hold for this next step, com-pleting the proof of Theorem 4.1. In Section 4.9, we retrospectively explain the purpose ofseveral objects and lemmas in this intricate proof.

Our notation will rarely distinguish between ZN (a probability space with the measure-preserving transformation τx = x + 1 mod N) and Z. Sets and functions on ZN willcorrespond to N -periodic sets and functions on Z, and any object on ZN is understood torepresent an object on ZMN for any M ∈ Z+.

Furthermore, we let P denote the uniform probability measure on ZN , and EX the ex-pected value of a random variable X : ZN → R. Note that the values of P and E are

CHAPTER 4. UNIVERSALLY L1-BAD ARITHMETIC SEQUENCES 33

unchanged when we consider a N -periodic set or function as an object on ZMN instead, sothat we may use P and E freely without keeping track of N . We will use X

d= Y to denote

that two random variables X and Y (not necessarily on the same probability space) haveidentical distributions.

Finally, we will use both subscripts and superscripts on certain functions fLh , gLh , X

Lh , Z

Lh

and certain sets Λγq ,Ξ

γq . To prevent these from being confused with exponential notation, we

note here that such superscripts on these objects will not denote exponents; we will thereforewrite the square of Xh as (Xh)

2 rather than X2h.

4.2 Outline of the Argument for the Squares

Here we will present a heuristic outline of the argument in the original case nk = k2,before introducing the necessary complications (exceptional sets and the like). We thereforeask the reader’s patience with these claims, some of which are not technically true; the ar-gument presented in Section 4.3 and thereafter will be rigorous.

Buczolich and Mauldin’s proof in [11] essentially boils down to three key insights. The firstis that in order to prove that k2 is universally L1-bad, it suffices to prove the existence ofwhat they term (K,M) families1 for arbitrary K,M ∈ N.

Definition Given K,M ∈ N and a measure-preserving system (Ω,F ,P, τ), a (K,M) familyon Ω consists of the following:

• f1, . . . , fK ∈ L1(Ω) with fh ≥ 0 and Efh ≤ 1

• X1, . . . , XK ∈ L1(Ω) pairwise independent with EXh ≥M and E(Xh)2 ≤ CM

• A measurable function Qx : Ω→ N such that for a.e. x ∈ Ω,

1

Qx

Qx∑k=1

fh(τk2

x) ≥ Xh(x) ∀1 ≤ h ≤ K (4.1)

Note that for each x ∈ Ω, Qx does not depend on h.

The point of constructing a (K,M) family is that, while a single Xh may have a weakL1 norm no greater than the L1 norm of f , an average of pairwise independent randomvariables with uniformly bounded variance is subject to the Weak Law of Large Numbers.Thus for some large K, the average X1(x)+···+XK(x)

Kwill be at least M

2on a set of probability at

1Buczolich and Mauldin define these families with many more conditions, in the fashion that we will defineour inductive Step (K,M,L) in Proposition 4.3; but the definition here will suffice for heuristic purposes.


least 12, giving a weak L1 norm of at least M

4, while the L1 norm of the average f1(x)+···+fK(x)

K

remains ≤ 1.

Therefore if we have (K,M) families for all K,M ∈ N, we can construct a dynamical systemand a function that violate any given weak (1,1) maximal inequality.

The second insight is that we can inductively construct a (K,M) family on the proba-bility space ZN with the shift operator τx = x+ 1 mod N , for some large N depending onK and M . We will use an inner induction: given a (K − 1,M) family, we will construct fKand XK in stages fLK , X

LK , so that we have all the required properties except for EXL

K ≥M ,which we replace with EXL+1

K ≥ EXLK + ε. Then for L large enough, we will have a (K,M)

family as desired.

The main obstruction to this approach is the difficulty of maintaining pairwise independence;when we alter fLK and XL

K , we must at the same time alter fL1 , . . . , fLK−1 and XL

1 , . . . , XLK−1

in order to sustain this property. We do this by taking into account properties of the distri-bution of the squares in residue classes.

Definition Given γ > 0, and given q ∈ N squarefree and odd with κ prime factors, considerthe following subsets of the integers:

Λq := x ∈ Z : (x, q) = 1, ∃k ∈ Z such that x ≡ k2 mod q(−Λq)

γ := −Λq + (0, γ2κ)

It follows from elementary number theory that every x ∈ Λq has exactly 2κ square rootsmodulo q; and that if q = p1 . . . pκ with pi large, then P(Λq) =

∏κi=1

pi−12pi≈ 2−κ. Clearly

P((−Λq)γ) ≤ γ; it follows from the result of Granville and Kurlberg [19] that P((−Λq)

γ) ≈1− e−γ.

We start with f 0K ≡ X0

K ≡ 1. At each step in L we have functions fL1 , . . . , fLK and

XL1 , . . . , X

LK , all periodic by some T = T (K,M,L). We will take some highly composite

q = q(K,M,L) T and set

fL+1K (x) := eγfLK(x)1Z\(−Λq)γ (x) (4.2)

XL+1K (x) := eγXL

K(x)1Z\(−Λq)γ (x) + c1Ψq(x), (4.3)

for some Ψq ⊂ (−Λq)γ with P(Ψq) ≥ cγ. Note that the L1 norm of fK is nearly unchanged

by this operation, but that EXL+1K = EXL

K + c2.

To maintain the property (4.1), for x 6∈ (−Λq)γ we will keep the same length of averag-

ing Qx we used for fLK ; but for x ∈ (−Λq)γ we will redefine Qx to be a multiple of q. We will


define the set Ψq such that if we add a generic square number to an element of Ψq, we havea good chance of landing in the support of fL+1

K ; the result of [19] ensures that this is possible.

We can ensure that XL+1K remains independent from our original (K−1,M) family by choos-

ing q relatively prime to T . However, we are not quite out of the woods: for x ∈ (−Λq)γ we

are now using different values of Qx with fL+1K than with our (K − 1,M) family.

This defect is repaired by the third insight of [11], the reason why we are working with(−Λq)

γ: if we replace all the other fLh with new functions supported near 0, the averagesover large Qx starting near any negative quadratic residue will hit this region frequently, andwe may engineer it so that those averages have the same distribution as the ones they arereplacing.

Using the inductive hypothesis, then, we construct a different (K − 1,M) family

gL1 , . . . , gLK−1, ZL1 , . . . , Z

LK−1, Q

′x,

where the lengths of averaging Q′x are relatively prime to Tq; we may then “restrict” it to(−Λq)

γ by taking

gLh (x) := q2−κgLh (x)1(0,γ2κ)+qZ(x)

ZLh (x) := ZL

h (x)1(−Λq)γ (x)

Q′x := qQ′x.

This preserves the property (4.1) on (−Λq)γ if we assume that

gLh (τQ′xy) = gLh (y +Q′x) = gLh (y) ∀y ∈ [x, x+ (qQ′x)

2), (4.4)

since if x ∈ (−Λq)γ, then |k ∈ [1, qQ′x] : x + k2 ∈ (0, γ2κ) + qZ| ≥ 2κQ′x and this set is

equidistributed modulo Q′x. This implies that for x ∈ (−Λq)γ,

1

Q′x

Q′x∑k=1

gLh (x+ k2) =1

qQ′x

Q′x∑k=1

2κ · q2−κgLh (x+ k2) ≥ Zh(x) = Zh(x).

(That is, averages of gLh over long intervals of the squares look like averages of gLh on shortintervals of the squares.) Then if for h ≤ K − 1 we let

fL+1h (x) := fLh (x)1Z\(−Λq)γ (x) + gLh

XL+1h (x) := XL

h (x)1Z\(−Λq)γ (x) + ZLh

and set the new Qx to equal Q′x on the set (−Λq)γ, we find that we have nearly preserved

the properties of the family we began with. Thus we may iterate the inner inductive step(to which we must of course add a version of (4.4)) and thereby construct a (K,M) family.


4.3 Main Theorem

Definition For a set Λ ⊂ Zt, we can define

P(Λ) :=|Λ|t

st := P(Λ)−1

Λγ := Λ + (0, γst) ⊂ Zt.

Remark Note that this turns Zt into a probability space, that st is the average spacingbetween elements of Λ in Zt, and that P(Λγ) ≤ γ. If Λ ⊂ Z is periodic by t, we can considerit as a subset of Znt for any n ∈ Z+, and we see that P(Λ) is independent of n.

The main result of this chapter is the following:

Theorem 4.1. Let nk ⊂ N be an increasing sequence, and α, β > 0. Say that for everyinteger t > 1, there exists a set of residues Λt ⊂ nk + tZ : k ∈ N ⊂ Zt such thatΛst = Λs ∩ Λt whenever (s, t) = 1, and that there exist some auxiliary sequences pj, qjof pairwise relatively prime positive integers such that

P(Λqj)→ 0, (4.5)

infj

P(Λpj) > 0, (4.6)

εγ := γ − lim infj→∞

P(Λγqj

) = o(γ), (4.7)

such that for all γ > 0 sufficiently small,

lim infj→∞

∣∣(u, v) ∈ Λqj × Λqj : |u− v − w| > γsqj ∀w ∈ Λqj

∣∣ > 5α|Λqj |2, (4.8)

and for all Q = pi1 . . . pikqj1 . . . qjl with i1 < · · · < ik, j1 < · · · < jl,

lim infN→∞

1

N|1 ≤ k ≤ N : nk ≡ a mod Q| > β

|ΛQ|∀a ∈ ΛQ. (4.9)

Then, given any C > 0 and any infinite set S ⊂ N, there exists a probability space (X,F ,P),a measure-preserving transformation τ on X, and an f ∈ L1(X) such that

‖ supN∈S| 1N

N∑k=1

f τnk |‖1,∞ > C‖f‖1.

Remark (4.7) states that Λqj does not cluster too much in Zqj so that P(Λγqj

) is nearlyγ, while (4.8) states that the set of differences of elements in Λqj is not concentrated nearΛqj . (4.9) states that each point of ΛQ is hit uniformly often by the sequence nk, if Q isany squarefree product of terms from the auxiliary sequences pj and qj. (The functionx→ Qx we will later construct will take such products as its values.)


Claim. Theorem 4.1 implies Theorem 1.4.

Proof. For the sequence of primes, we take Λt to be the set of integers relatively prime tot, then let pj be distinct primes and qj be highly composite such that q−1

j φ(qj) → 0; since∏p prime

p− 1

p= 0, it is possible to choose such qj. The property (4.9) is clear from Dirichlet’s

theorem on arithmetic progressions.

For the sequence of dth powers, we will take Λt to be the residues of kd mod t which are unitsmod t. We will take pj to be distinct primes congruent to 1 mod d, and qj to be productsof j such primes. Note that if Q has κ prime factors each congruent to 1 mod d, then everyx ∈ ΛQ will have precisely dκ dth roots in ZQ, and P(ΛQ) = φ(Q)d−j. If we choose all of theprime factors sufficiently large, we can ensure that φ(Q) ≥ 1

2for all squarefree products Q

of these sequences, so that (4.9) is satisfied.

To prove (4.7) and (4.8) for each of these cases, we will use some recent results on thedistribution of the residues Λqj (as the average spacing sqj → ∞); roughly speaking, ineach case they are distributed locally like a Poisson process of rate P(Λqj). We begin byintroducing some notation.

Definition Let ykNk=1 be a strictly increasing sequence of real numbers in [0, N), andconsider it as a subset of T = R/NZ. For E ⊂ 1, . . . , N, and θ > 0, we define a probabilitymeasure

Pk(E) :=|E|N

(4.10)

and a cumulative distribution function

F (θ) := Pk(|yk+1 − yk| > θ). (4.11)

Now if we consider the normalized set s−1q Λq = yi : 1 ≤ i ≤ |Λq| ⊂ [0, |Λq|) (where the yi

are taken in increasing order), we can examine the cumulative distribution function Fq(θ)defined as above. Hooley [20] proves for the primes, and Granville and Kurlberg [19] provefor the dth powers2, that Fq(θ)→ e−θ pointwise as sq →∞ (for which reason they call thesesets “Poisson distributed”). Thus

P(Λγq ) = |Λq|−1

∑i

|yi+1 − yi| ∧ γ =

∫ γ

0

Fq(θ)dθ → 1− e−γ,

so that we have (4.7). To prove (4.8) from this distributional fact, however, requires a littlemore work.

2While Corollary 2 in that paper is stated for the entire set of dth power residues modulo q, we see thatpassing to the subset Λq does not change equation (1) and thus we may apply the same result.


Now if γ < 112

, then e−2γ > 67

and we can set J := log 2γ

> 4; then Lemma 4.2 implies

Pl(ζl > 1/8) ≥ Pl(ζl >

−1 + e−Jγ + e−2γ

2+ oj(1)

)≥ e−2γ − e(2−J)γ + oj(1) ≥ 1

4

for j sufficiently large, so that (4.8) holds with α = 1160

.

Remark Note that (4.9) cannot be satisfied by polynomials other than nk = cdkd + c0, as

can be seen from Cohen’s result [16] that if we fix a ∈ Zp[x] of degree d which is not of thistype and consider a(x)− y as y ∈ Zp varies, for some fixed proportion of y this polynomialwill have d distinct roots, while for some fixed proportion it will have 1 root. This prevents(4.9) from holding for sufficiently composite products. However, the author expects thatevery polynomial of degree 2 or greater over Z should be persistently universally L1-bad,and that some clever variant of this argument should suffice to prove as much.

4.4 The Inductive Step

For an inductive argument to work, we will have to specify additional properties of theobjects we seek, including a fixed distribution for the functions XL

h .

Definition Given the sequences pj and qj as in Theorem 4.1, denote the set of their(squarefree) products

Q := pi1 . . . pikqj1 . . . qjl : i1 < · · · < ik, j1 < · · · < jl (4.12)

and the functions (depending on the infinite S ⊂ N in Theorem 4.1)

N(Q) := max

N : ∃a ∈ ΛQ : |1 ≤ k ≤ N : nk ≡ a mod Q| ≤ βN

|ΛQ|

(4.13)

ψ(n) := infs ∈ S : s > N(Q) ∀Q ∈ Q, Q ≤ n. (4.14)

Definition For 0 < γ < 1 and 0 < α < 1, we recursively define functions Yn,γ,α : [0, 1]n+1 →R, which we may consider as random variables with the Lebesgue measure. Let Y0,γ,α(x0) ≡1, and

Yn+1,γ,α(x0, . . . , xn+1) := (1− γ)−1Yn,γ,α(x0, . . . , xn)1[γ,1)(xn+1) + α1[0,αγ)(xn+1). (4.15)

Note that

E(Yn,γ,α) = EYn−1,γ,α + α2γ = 1 + nα2γ,

E(Yn,γ,α)2 = (1− γ)−1E(Yn−1,γ,α)2 + α3γ ≤ (1− γ)−n(1 + α3) ≤ 2(1− γ)−n.


Given the conditions of Theorem 4.1, we may assume that all the pj and qj are odd and thatα is a dyadic rational. Choose γ0 < 1/2 such that for all 0 < γ < γ0, (4.7) and (4.8) holdand εγ < αγ; we will write ε = εγ from now on unless otherwise specified. We are now readyto state the inductive step:

Proposition 4.3 (Step (K,M,L)). Assume the conditions of Theorem 4.1. Given anydyadic rational 0 < γ < γ0, any K,M,L ∈ N with L ≤ M , constants A, δ > 0 with δ < 8ε,and an odd integer D, there exist T ∈ Q and R ∈ N with (T,D) = (R,D) = 1 and thefollowing objects:

(1) f1, . . . , fK ∈ `1(ZT ) with fh ≥ 0 and 1 ≤ Efh ≤ (1 + 4ε)(K−1)M+L.

(2) X1, . . . , XK ∈ `1(ZRT ) pairwise independent with Xhd= YM,γ,α ∀h < K and XK

d= YL,γ,α

(3) An exceptional set E ⊂ ZT with P(E) ≤ δ

(4) Qx : ZT → Z+ with Qx ∈ Q and Qx | T ∀x, such that for each x 6∈ E,

1

|ΛQx|∑

a∈ΛQx ,1≤a≤Qx

fh(x+ a) ≥ Xh(x) ∀1 ≤ h ≤ K (4.16)

and

fh(x+ y −Qx) = fh(x+ y) ∀1 ≤ h ≤ K, Qx ≤ y ≤ ψ(AQx). (4.17)

Remark In the final section of this paper, we will discuss the significance of the parametersA,D,R, and δ, as well as the distribution Yn,γ,α, the reason we require γ and α to be dyadicrationals, and other points whose necessity in the argument is not immediately obvious. Forthe time being, we ask the reader’s trust that these complications are required in order tomake a strong enough inductive step.

Claim. Proposition 4.3 implies Theorem 4.1.

Proof. If we fix C > 0, take γ > 0 small, and set M = bC/γc and K = bγ/Cεc, and takeδ ≤ 1

4, A = D = 1, then at Step (K,M,M) we have

Efh ≤ (1 + 4ε)KM ≤ (1 + 4ε)1/ε ≤ e4,

EXh = EYM,γ,α ≥ 1 +Mα2γ ≥ Cα2,

E(Xh)2 = EY 2

M,γ,α ≤ 2(1− γ)−M ≤ 2e2γM ≤ 2e2C .


Therefore, since the Xh are pairwise independent and identically distributed,

P(X1 + · · ·+XK

K≤ Cα2/2

)≤ P

(∣∣∣∣X1 + · · ·+XK

K− EX1

∣∣∣∣ ≥ Cα2/2

)≤ (Cα2/2)−2E

∣∣∣∣X1 + · · ·+XK

K− EX1

∣∣∣∣2≤ 4α−2C−2K−1E(X1 − EX1)2

≤ 8e2C

α2C2K

and for γ sufficiently small (by (4.7), this means K = bγ/Cεc sufficiently large), this is lessthan 1

2.

Now if we consider the probability space ZRT with the measure-preserving transformationτx = x + 1, and set f = f1 + · · · + fK and X = X1 + · · · + XK , then by (4.9), (4.16) and(4.17)

1

ψ(Qx)

ψ(Qx)∑k=1

f(x+ nk) >β

|ΛQx|∑a∈ΛQx

f(x+ a) ≥ βX(x) ∀x 6∈ E, (4.18)

and since ψ(Qx) ∈ S, this implies

P

(supN∈S

1

N

N∑k=1

f τnk > CKα2β/2

)≥ P

(X

K>Cα2

2

)− P(E) ≥ 1

4.

Therefore

‖ supN∈S

1

N

N∑k=1

f τnk‖1,∞ ≥CKα2β

8≥ Cα2β

8e4‖f‖1.

Since C is arbitrary, there can be no maximal inequality.

We will prove Proposition 4.3 by induction on K and L. We fix M and γ at the beginning ofthe argument (since they will not change as K and L change), and prove each Step (K,M,L)for all values of A, δ and D.

Note that Step (1,M, 0) is trivial, and that Step (K − 1,M,M) implies Step (K,M, 0):fix the parameters A, δ and D and obtain f1, . . . , fK−1, X1, . . . , XK−1, E,Qx satisfying(1)-(4) on ZT . Now set fK ≡ 1, XK ≡ 1 on Z. This clearly satisfies the conditions.

Therefore, to prove Theorem 4.1, it suffices to show that if L < M and we know Step(K,M,L) and all previous steps (for all values of A, δ and D, and for a fixed γ), we can prove


Step (K,M,L+1) for a fixed A, δ andD and the same γ. We start by applying Step (K,M,L)with A, δ/4 and D to obtain a family TL, RL, f

L1 , . . . , f

LK , X

L1 , . . . , X

LK , EL, Qx,L satisfying

(1)-(4); we must then construct TL+1, RL+1, fL+11 , . . . , fL+1

K , XL+11 , . . . , XL+1

K , EL+1, Qx,L+1to have the required properties.

4.5 Periodic Rearrangements

As in [11], we will essentially construct the next functions by choosing a new q and redefin-ing the current functions on the subset (−Λq)

γ. On this set, we will be selecting a newQx,L+1 TL, and we want to ensure that the left side of (4.16) will be uniformly large ona significant subset of (−Λq)

γ. Since an average of fLh over ΛQ (in the sense of (4.16)) maybe irregular for large Q, we will modify the fLh in advance so that these averages will bebounded below by a constant, while preserving their averages over ΛQ for smaller Q.

Following Buczolich and Mauldin, we call this modification a periodic rearrangement. Givennatural numbers p T with (p, T ) = 1, we will define a linear operator f → f from `1(ZT )to `1(ZpT ) which preserves joint distribution of functions, such that on long blocks each f isidentical to a translate of f .

In our particular cases, where ΛT consists of the residues of dth powers or the integersrelatively prime to T , we can simply define

f(x) :=

f(y) x ∈ y + pZ, 0 ≤ y < T · bp/T cf(x) otherwise,

and prove directly that for every x ∈ ZpT ,

1

|ΛpT |∑a∈ΛpT

f(x+ a) ≥ 1

2Ef.

(For the former case, we would use the Polya-Vinogradov inequality on character sums; forthe latter we would use Dirichlet’s theorem.)

However, we lack such tools when considering more general sequences, so we shall instead usean external randomization in our construction of f . Let Ω be a probability space and ξi(ω)be independent random variables on Ω, each with a uniform distribution on the discrete set0, . . . , T − 1. Then for any f ∈ `1(ZT ) and ω ∈ Ω we define (on the interval [0, pT )) thefunction

fω(x) :=

f(x+ ξi(ω)) (i− 1)b√pcT ≤ x ≤ ib√pcT, 1 ≤ i ≤ √p;f(x) (b√pc)2T ≤ x < pT.

(4.19)


Heuristically speaking, we break ZpT into blocks of size T b√pc and shift each by a randomvariable ξi(ω). The point of this is that, although we cannot directly prove that ΛpT will beequidistributed in the residue classes modulo T , there must exist some value of these shiftsunder which this is approximately so.

Similarly, for a set E ⊂ ZT we can define Eω := supp 1ωE ⊂ ZpT . Note that Eω con-tains exactly b√pc|E| points in each full block, and (p− (b√pc)2)|E| points in the last block;

thus we see that P(Eω) = P(E) for all E ⊂ ZT and all ω ∈ Ω, from which it follows thatthis periodic rearrangement preserves the joint distribution of any collection of functions.

For most x ∈ ZpT there exists an x such that fω(x + z) = f(x + z) for 0 ≤ z √p.In particular, if we take our original exceptional set E ⊂ ZT , we can define a new excep-tional set

E1(ω) := Eω ∪⋃

0≤i≤√p

T−1

[iT b√pc − ψ(AT ), iT b√pc] + pZ (4.20)

such that (4.16) and (4.17) are still satisfied for fL,ωh , XL,ωh and Qω

x,L off of E1(ω). Further-more,

P(E1(ω)) ≤ P(E) +ψ(AT ) + 2T

T√p

≤ δ/2

for p sufficiently large.

Now we can prove the following lemma:

Lemma 4.4. For p sufficiently large with (p, T ) = 1, there exists ω ∈ Ω such that for all0 ≤ f ∈ `1(ZT ) and all x ∈ ZpT ,

1

|ΛpT |∑a∈ΛpT

fω(x+ a) ≥ 1

2Ef.

Proof. By the linearity of the periodic rearrangement, it suffices to prove that for some ω,this holds for all characteristic functions of singletons in ZT : thus it is enough to show thatfor all x ∈ ZpT and b ∈ ZT ,

Pω

∑a∈ΛpT

1ωb(x+ a) <|ΛpT |2T

<1

pT 2.


Fix b and x; then counting only the main blocks,∑a∈ΛpT

1ωb(x+ a) ≥∑

1≤i≤√p

∣∣∣∣a ∈ ΛpT :(i− 1)b√pcT ≤ x+ a < ib√pcT,x+ a+ ξi(ω)− b ∈ TZ

∣∣∣∣=

∑i

|ΛpT ∩ Pi(ω)|,

where Pi(ω) :=

[(i− 1)b√pcT − x, ib√pcT − x) ∩ b− x− ξi + TZ

is an arithmetic pro-gression in ZpT . If we then define

νi(ω) := p−1/2|ΛpT ∩ Pi(ω)|,

we see that 0 ≤ νi ≤ 1, that the νi are independent, and that each z ∈ ΛpT ∩ [(i−1)b√pcT −x, ib√pcT − x) contributes to νi(ω) for precisely one value of ξi(ω). Therefore

Eωνi(ω) =1

T√p|ΛpT ∩ [(i− 1)b√pcT − x, ib√pcT − x)|,

Var ωνi(ω) ≤ Eων2i (ω) ≤ Eωνi(ω).

Recall that infj

P(Λpj) > 0, so that for p T , |ΛpT | = |Λp| · |ΛT | T√p. Thus for p

sufficiently large,

Eω(∑i

νi(ω)) ≥ 1

T√p|ΛpT \ [x− T (p− (b√pc)2), x]| > 3|ΛpT |

4T√p

Var ω(∑i

νi(ω)) ≤ |ΛTp|T√p

and we may apply Chernoff’s Inequality (Theorem 1.8 from [29]) to find

Pω

∑a∈ΛpT

1ωb(x+ a) <|ΛpT |2T

≤ Pω

(∑i

|ΛpT ∩ Pi(ω)| < |ΛpT |2T

)

= Pω

(∑i

νi(ω) <|ΛpT |2T√p

)

≤ Pω

(∑i

νi(ω)− Eωνi >|ΛpT |4T√p

)

≤ 2 exp(− |ΛpT |64T√p

) <1

pT 2

for p sufficiently large (depending on T ).


4.6 Defining fL+1K

Using the properties (4.5)-(4.8), we now take p = pi(K,L) and q = qj(K,L) from our auxiliarysequences such that they are relatively prime to each other and to TL, RL and D, such thatp is large enough for Lemma 4.4 and such that q D, sq > 8δ−1ψ(AT ) and

P((−Λq)γ) = P(Λγ

q ) ≥ γ − ε,|(u, v) ∈ Λq × Λq : |u− v − w| > γsq ∀w ∈ Λq| ≥ 5α|Λq|2.

We would like to define fL+1K and XL+1

K as in (4.2) and (4.3), but then XL+1K will not pre-

cisely equal YL+1,γ,α in distribution. This is the reason we will make the XL+1h periodic by

RL+1TL+1 rather than just TL+1: we will later multiply the parts of XL+1K by the characteris-

tic function of intervals whose lengths are appropriate multiples of TL+1. It will be essential(for its use in later inductive steps) that we keep (RL+1, D) = 1, and for this we will needto define fL+1

K and XL+1K in a more complicated fashion.

First, we will let

Φq := u ∈ −Λq : |v ∈ Λq : u+ v /∈ −Λq + (−γsq, γsq)| ≥ 2α|Λq| (4.21)

and note that |Φq| ≥ 3α|Λq|. We then consider Φγq = Φq + (0, sqγ) ⊂ (−Λq)

γ. Now P(Φγq ) ≥

2αγ since

γ − ε ≤ P((−Λq)γ) ≤ P(Φγ

q ) + q−1γsq| − Λq \ Φq| ≤ P(Φγq ) + γ(1− 3α)

and we have stipulated that ε ≤ αγ.

Now we choose two sets Ψq,∆q ⊂ Zq such that

Ψq ⊂ Φγq (4.22)

P(Ψq) ≥ αγ (4.23)

Zq \ (−Λq)γ ⊂ ∆q ⊂ Zq \ Φq (4.24)

P(∆q) < 1− P(Λγq ) + δ/8 (4.25)

(|Ψq|, D) = (|∆q|, D) = 1. (4.26)

(As q D, this last condition is clearly possible to satisfy simultaneously with the others.)We observe that

|v ∈ Λq : x+ v ∈ ∆q| ≥ 2α|Λq| ∀x ∈ Ψq, (4.27)

and accordingly we define

fL+1K := (1− γ)−1fLK1∆q ∈ `1(ZqpT ).


Note that (q, pT ) = 1 implies EfL+1K = (1 − γ)−1P(∆q)EfLK = (1 − γ)−1P(∆q)EfLK ; since

γ < 1/2 and δ < 8ε, this means

EfLK ≤ EfL+1K ≤ (1 + 4ε)EfLK . (4.28)

The goal of Lemma 4.4 and the definition of Ψq is the following lemma:

Lemma 4.5. Let T = TL, with p and q chosen as above. For all x ∈ Ψq and for any Q ∈ Qsuch that qpT | Q,

1

|ΛQ|∑a∈ΛQ

fL+1K (x+ a) ≥ α.

Proof. Let B := QqpT

. Since Q ∈ Q is squarefree, we see that B, T, p and q are pairwiserelatively prime, and

1

|ΛQ|∑a∈ΛQ

fL+1K (x+ a) =

1

|ΛB|∑w∈ΛB

1

|ΛqpT |∑

z∈ΛqpT

fL+1K (x+ z)

=1

|ΛpT |1

|Λq|∑u∈ΛpT

∑v∈Λq

(1− γ)−1fLK(x+ u)1∆q(x+ v)

≥ 2α

|ΛpT |∑u∈ΛpT

fLK(x+ u)

≥ αEfLK ≥ α

using (4.27) for the first inequality and Lemma 4.4 for the second.

We also define an additional exceptional set

E2L := x ∈ ∆q : ∃0 < y ≤ ψ(AQx,L) such that x+ y /∈ ∆q. (4.29)

Note that E2L ⊂ ∆q ∩ (−Λq)

γ ∪ −Λq + (−ψ(AT ), 0]. By our choice of q, we see that

P(E2L) ≤ δ/8 + |Λq|ψ(AT ) ≤ δ

4.

4.7 Restricting a Family to (−Λq)γ

As noted in the heuristic outline, if we wish to change Qx on the set (−Λq)γ, we must change

fL1 , . . . , fLK−1 and XL

1 , . . . , XLK−1 as well, since these need no longer satisfy (4) with the new

value of Qx. In order to find suitable replacement functions on (−Λq)γ, we will use take a

Step (K − 1,M,M) family S,R′, g1, . . . , gK−1, Z1, . . . , ZK−1, E′, Q′x with suitable param-

eters, and then restrict the functions gh to the set (0, γsq) + qZ (multiplying them by |Λq|


so that their `1 norm is γ times its previous value). The averages along Λq starting at anyx ∈ (−Λq)

γ will sample from this set uniformly often, so that we may be able to preserveproperty (4) there.

This restriction will not preserve the independence of the Zh as such, since the averagesof the restricted gh must be 0 off of (−Λq)

γ. However, we may define the conditional expec-tations E(X | Σ); and these will remain independent, for Σ ∈ (−Λq)

γ.

Definition Let X be a random variable on a discrete probability space (Ω,P), and let Σ ⊂ Ωwith P(Σ) > 0. We define the conditional expectation E(X | Σ) to be the random variableon Σ with E(X | Σ)(x) = X(x) ∀x ∈ Σ, where Σ is equipped with the probability measurePΣ(x) = P(Σ)−1P(x).

Now we can state the actual form of this restriction for an entire Step (K,M,L) family:

Lemma 4.6. Let T,R, f1, . . . , fK , X1, . . . , XK , E,Qx satisfy the properties of Step (K,M,L)for the parameters A, δ,D. Say we have q, B ∈ Q with q, B and T pairwise relatively prime,and qB ≤ A. Let

Ξγq := (0, γsq) + qZ

fh(x) := |Λq|fh(x)1Ξγq (x)

Xh(x) := Xh(x)1(−Λq)γ (x)

E := E ∩ (−Λq)γ

Qx := qBQx

A :=A

qB

D :=D

(D, qB)

T := qBT.

Then (T , D) = 1 and

(1) f1, . . . , fK ∈ `1(ZT ) with fh ≥ 0 and Efh ≤ γEfh

(2) X1, . . . , XK ∈ `1(ZRT ) such that for any nonempty q-periodic Σ ⊂ (−Λq)γ, E(X1 |

Σ), . . . ,E(XK | Σ) are independent and E(Xh | Σ)d= Xh.

(3) E ⊂ ZT with P(E) ≤ δP((−Λq)γ)

(4) Qx : ZT → N such that Qx | T ∀x ∈ ZT and for each x 6∈ E,

1

|ΛQx|∑a∈ΛQx

fh(x+ a) ≥ Xh(x) ∀1 ≤ h ≤ K


and

fh(x+ y −Qx) = fh(x+ y) ∀1 ≤ h ≤ K, Qx ≤ y ≤ ψ(AQx).

Proof. Since (q, T ) = 1, any q-periodic set contains an equal portion of integers from eachresidue class modulo T . This fact quickly implies properties (1)-(3), noting that the jointdistribution of the Xh on any q-periodic Σ ⊂ (−Λq)

γ is the same as their joint distributionon ZRT . Now for property (4), take x ∈ (−Λq)

γ \ E (since it is trivial otherwise). Since Qx,q, and B are relatively prime,

1

|ΛQx|∑a∈ΛQx

fh(x+ a) =1

|ΛB|∑u∈ΛB

1

|ΛqQx|∑

z∈ΛqQx

fh(x+ z)

=1

|Λq|∑v∈Λq

|Λq|1Ξγq (x+ v) · 1

|ΛQx|∑

w∈ΛQx

fh(x+ w)

≥ 1

|ΛQx|∑

w∈ΛQx

fh(x+ w) ≥ Xh(x) = Xh(x)

since if x ∈ (−Λq)γ, there must exist some v ∈ Λq such that x+ v ∈ Ξγ

q .

Finally, AQx = AQx so (4.17) implies the last claim trivially.

4.8 Completion of the Inductive Step

Now we are ready to define the other functions and prove Step (K,M,L + 1); but since somuch goes into this step, we will show the origins of the various pieces.

We began with TL, RL, fL1 , . . . , f

LK , X

L1 , . . . , X

LK , EL, Qx,L satisfying (1)-(4) with the pa-

rameters A, δ/4, and D. We modify these objects in several ways, using a new p = pi(K,L)

and q = qj(K,L) chosen in Section 6.

First, we applied the p-periodic rearrangement f → fω defined in (4.19) to the functions fLhand XL

h , with ω chosen as in Lemma 4.4, and defined an associated exceptional set E1L(ω)

in (4.20). We defined sets ∆q,Ψq ⊂ Zq satisfying (4.22)-(4.27), then we defined the functionfL+1K in (4.28) and an additional exceptional set E2

L in (4.29).

We are now ready to proceed.

Set AL := ATLpq and DL := DTLpq. By our strong inductive hypothesis, we may assumeStep (K − 1,M,M) and thus construct a family S,R′, g1, . . . , gK−1, Z1, . . . , ZK−1, E

′, Q′x


satisfying (1)-(4) with parameters AL, δ/4, DL. (In particular, S and R′ are each relativelyprime to TL, p, q, and D.)

Applying Lemma 4.6 with B = Tp, we obtain S, R′, g1, . . . , gK−1, Z1, . . . , ZK−1, E′, Q′x

on (−Λq)γ satisfying (1)-(4) with the parameters AL = A, δ/4, and DL = D.

Then we define

TL+1 := S = STLpq (4.30)

EL+1 := E1L(ω) ∪ E2

L ∪ E ′ (4.31)

Qx,L+1 :=

Qx,L, x ∈ ∆q

Q′x, x 6∈ ∆q,(4.32)

and for 1 ≤ h ≤ K − 1,

fL+1h := fLh 1∆q + gh1Z\∆q ∈ `1(ZTL+1

) (4.33)

XL+1h := XL

h 1∆q + Zh1Z\∆q ∈ `1(ZTL+1). (4.34)

We have already defined

fL+1K := (1− γ)−1fLK1∆q ∈ `1(ZTL+1

).

It thus remains to define XL+1K . As noted before, we cannot simply define it as in (4.3); we

must reduce it slightly so that it equals YL+1,γ,α in distribution.

Recall that P(∆q) ≥ 1 − γ and P(Ψq) > αγ; we have taken γ, α to be dyadic rationalsand assumed (4.26), so we may write

1− γP(∆q)

=s

r,

αγ

P(Ψq)=t

r(4.35)

with (r,D) = 1. (Recall that D is odd.) Everything so far is periodic with period TL+1RLR′,

so if we define

RL+1 := RLR′r

Γs := [0, sR′RLTL+1) +RL+1TL+1ZΓt := [0, tR′RLTL+1) +RL+1TL+1Z

XL+1K := (1− γ)−1XL

K1∆q1Γs + α1Ψq1Γt ,

then XL+1K

d= YL+1,γ,α and all of the XL+1

h are periodic with period RL+1TL+1.

We now have a family TL+1, RL+1, fL+11 , . . . , fL+1

K , XL+11 , . . . , XL+1

K , EL+1, Qx,L+1, with TL+1 ∈Q and (TL+1, D) = (RL+1, D) = 1. We must check the four properties of Step (K,M,L+ 1):


(1) fL+11 , . . . , fL+1

K ∈ `1(ZTL+1) with fL+1

h ≥ 0 and 1 ≤ EfL+1h ≤ (1 + 4ε)(K−1)M+L+1.

Proof. For 1 ≤ h ≤ K − 1, by the inductive hypothesis we see

EfL+1h = EfLh P(∆q) + EgLh

≤ (1− γ + ε)EfLh + γEgLh≤ (1 + ε)(1 + 4ε)(K−1)M+L

≤ (1 + 4ε)(K−1)M+L+1.

For fL+1K , this follows from (4.28) and the inductive hypothesis.

(2) XL+11 , . . . , XL+1

K ∈ `1(ZRL+1TL+1) pairwise independent with XL+1

h

d= YM,γ,α for 1 ≤ h ≤

K − 1, and XL+1K

d= YL+1,γ,α.

Proof. We have inductively assumed that XL1 , . . . , X

LK are pairwise independent; since

the periodic rearrangement preserves joint distribution, this is true of XL1 , . . . , X

LK as

well. We begin by considering the conditional expectations of the XLh on ∆q. Recall

that q is relatively prime to p, TL, RL, S, and R′, so that ∆q contains an equalproportion of all residue classes modulo pTLRLSR

′. Furthermore, Γs and Γt eachcontain an equal proportion of all residue classes modulo R′RLTL+1. Thus for anyh < K and λh, λK > 0,

P(x ∈ ∆q : XL+1h (x) ≥ λh, X

L+1K (x) ≥ λK)

= P(x ∈ ∆q ∩ Γs, XLh (x) ≥ λh, X

LK(x) ≥ λK)

=s

rP(∆q)P(XL

h (x) ≥ λh)P(XLK(x) ≥ λK)

= P(∆q)P(XL+1h (x) ≥ λh)P(XL+1

K (x) ≥ λK).

Thus E(XL+1h | ∆q) and E(XL+1

K | ∆q) are independent; similarly, E(XL+1h | ∆q) and

E(XL+1h′ | ∆q) are independent for any h < h′ < K.

We proceed similarly on the rest of Z, letting Σ denote either Ψq or ZTL+1\ (∆q ∪Ψq);

on each of these, Lemma 4.6 implies that E(ZL1 | Σ), . . . ,E(ZL

K−1 | Σ) are pairwiseindependent and distributed like YM,γ,α. Clearly E(1Γt | Σ) is independent of anyof these, so E(XL+1

1 | Σ), . . . ,E(XL+1K | Σ) are pairwise independent for Σ = Ψq or

ZTL+1\ (∆q ∪Ψq).

Now for each 1 ≤ h < K,

E(XL+1h | ∆q)

d= E(XL+1

h | Ψq)d= E(XL+1

h | ZTL+1\ (∆q ∪Ψq))

d= XL+1

h

d= YM,γ,α, (4.36)


and so for any 1 ≤ h < h′ ≤ K and λh, λh′ > 0,

P(XL+1h (x) ≥ λh, X

L+1h′ (x) ≥ λh′)

=∑

Σ

P(x ∈ Σ : XL+1h (x) ≥ λh, X

L+1h′ (x) ≥ λh′)

=∑

Σ

P(E(XL+1h | Σ)(x) ≥ λh,E(XL+1

h′ | Σ)(x) ≥ λh′)P(Σ)

=∑

Σ

P(E(XL+1h | Σ)(x) ≥ λh)P(E(XL+1

h′ | Σ)(x) ≥ λh′)P(Σ)

=∑

Σ

P(XL+1h (x) ≥ λh)P(E(XL+1

h′ | Σ)(x) ≥ λh′)P(Σ)

= P(XL+1h (x) ≥ λh)

∑Σ

P(x ∈ Σ : XL+1h′ (x) ≥ λh′)

= P(XL+1h (x) ≥ λh)P(XL+1

h′ (x) ≥ λh′).

(The sum is over the sets Σ = ∆q,Ψq, and ZTL+1\ (∆q∪Ψq); the property (4.36) enters

in at the fourth equality.) Thus we have preserved independence.

We have already noted that XL+1K

d= YL+1,γ,α.

(3) An exceptional set EL+1 ⊂ ZTL+1with P(EL+1) ≤ δ.

Proof. P(EL+1) ≤ P(E1L(ω)) + P(E2

L) + P(E ′L) ≤ δ/2 + δ/4 + δγ/4 ≤ δ.

(4) Qx = Qx,L+1 : ZTL+1→ Z+ with Qx ∈ Q and Qx | TL+1 ∀x, such that for each x 6∈ EL+1,

1

|ΛQx|∑a∈ΛQx

fL+1h (x+ a) ≥ XL+1

h (x) ∀1 ≤ h ≤ K

and

fh(x+ y −Qx) = fh(x+ y) ∀1 ≤ h ≤ K, Qx ≤ y ≤ ψ(AQx).

Proof. Since Qx,L | TL and Q′x | S, clearly Qx,L+1 | TL+1; in addition, Qx,L ∈ Q andQ′x = pqTLQ

′x ∈ Q (this is squarefree since p, q, TL, and Q′x are pairwise relatively

prime).

For x ∈ ∆q \ EL+1 and 1 ≤ y ≤ ψ(ATL), we see that

fL+1h (x+ y) = fLh (x+ y) = fLh (x+ y + ξi(ω))


for some ω fixed and i depending only on x. Thus (4) follows from the previous step.

On Ψq and Z \ (∆q ∪ Ψq), since pqTL | Q′x, this is just Lemma 4.6 for h < K, andfor h = K this is Lemma 4.5 combined with the observation that fL+1

K is periodic bypqTL.

Thus we have proved Step (K,M,L+ 1); by induction, we have proved Proposition 4.3 andTheorem 4.1.

4.9 Notes on the Proof

Several of the conditions, parameters and lemmas in this complicated argument appearon a first reading to be extraneous to the proof. In the interest of clarity, we find it helpfulto outline in hindsight the purposes of the following:

• The condition (4.7) lets us prove Theorem 4.1 from Proposition 4.3 (note that thisproof requires γ/Cεγ → ∞ as γ → 0). (4.8) comes in at (4.21) and the subsequentdefinition of Ψq, and (4.9) allows us to claim (4.18).

• Prescribing an exact distribution Yn,γ,α (defined in (4.15)) for the XLh is necessary in

order to guarantee (4.36), which ensures that pairwise independence is preserved.

• γ and α must be dyadic rationals, and ∆q and Ψq must be chosen to satisfy (4.26),so that we can assume (r,D) = 1 in (4.35), so that we can have (R,D) = 1 in Step(K,M,L), so that we can choose our (K − 1,M,M) family with (R′, q) = 1, so thatXL+1K will be independent of the other XL+1

h .

• The parameter D lets us guarantee that when we inductively introduce a Step (K −1,M,M) family, we can ensure that its period R′S is relatively prime to TL, p, and q,thus allowing us to apply Lemma 4.6.

• We have two distinct parameters T and R because we will need the period of fLK to besquarefree in Lemma 4.5 (because the result of [19] only applies for squarefree moduli),but the operation of reducing XL+1

K to its proper distribution will multiply its periodby a large power of 2.

• The condition (4.17) is necessary for (4.18), connecting the actual averages over thesequence nk with the averages over a set of residues |ΛQ| ∈ ZQ. The parameter Amust be allowed to take arbitrarily large values, although it need only be ≥ 1 whenused in (4.18), because each application of Lemma 4.6 divides it by a large constant.


• The periodic rearrangement defined in (4.19) puts a uniform lower bound on the av-erages of fLK over Qx,L+1 on a set which depends only on q and not on fLK ; this allowsus to prove Lemma 4.5.

54

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