WACaM’04July 17, 2004 A. FridGrowth of arithmetical complexity1 Possible growth of arithmetical...
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Transcript of WACaM’04July 17, 2004 A. FridGrowth of arithmetical complexity1 Possible growth of arithmetical...
A. Frid Growth of arithmetical complexity1
WACaM’04 July 17, 2004
Possible growth of arithmetical complexity
Anna FridSobolev Institute of Mathematics
Novosibirsk, Russia
[email protected]://www.math.nsc.ru/LBRT/k4/Frid/fridanna.htm
A. Frid Growth of arithmetical complexity2
WACaM’04 July 17, 2004
Arithmetical closure
is the arithmetical closure of since
......21 nwwww
}0,0,|...{)( )1( ndkwwwwA dnkdkk
)(wA
).())(( wAwAA
w
A. Frid Growth of arithmetical complexity3
WACaM’04 July 17, 2004
Van der Waerden theorem
)(wA was invented by S. V. Avgustinovich in 1999
but
)(wAna
Theorem (Van der Waerden, 1927):
always contains arbitrarily long powers of some symbol .a
What else may occur in ?)(wA
A. Frid Growth of arithmetical complexity4
WACaM’04 July 17, 2004
A simple question
Does 010101… always occur in the Thue-Morse word?
0110 1001 1001 0110 1001 0110…
Proof:
Too precise question: in fact, any binary word does.
Avgustinovich, Fon-Der-Flaass, 1999
)()(, wAbawAba
A. Frid Growth of arithmetical complexity5
WACaM’04 July 17, 2004
Subword and arithmetical complexity
Subword complexity Arithmetical complexity)(nfw )(naw
)()( nanf ww nw,Growing functions
periodic ult. constant
non-periodic 1nW is complexity is
number of factors of w of length n
number of words of length n in A(w)
A. Frid Growth of arithmetical complexity6
WACaM’04 July 17, 2004
Possible growth?
How can complexity grow?subword
Many examples, no characterization
Is the question trivial? Maybe it is always exponential?
How can complexity grow?arithmetical
NO
A. Frid Growth of arithmetical complexity7
WACaM’04 July 17, 2004
Paperfolding word
P=0?1? – a pattern w=T(P,P,…)=T(P)
1 0 10 1 0 10w 0 01 110 0
aw(n)=8n+4 for n > 13
A generalization: Toeplitz words
the paperfolding word
A. Frid Growth of arithmetical complexity8
WACaM’04 July 17, 2004
First results and classification
Exponentialar. compl.
Linearar. compl.
Fixed pointsof uniform morphisms
Paperfolding wordThue-Morse word
[Avgustinovich, Fon-Der-Flaass, Frid, 00 (03)]
A. Frid Growth of arithmetical complexity9
WACaM’04 July 17, 2004
Arguments for arithmetical complexity
Mathematics involved:
• Van der Waerden theorem
• more number theory: Legendre symbol, Dirichlet theorem, computations modulo p… (for words of linear complexity)
• linear algebra (for the Thue-Morse word etc.)
• geometry (for Sturmian words)
• …
A. Frid Growth of arithmetical complexity10
WACaM’04 July 17, 2004
Further results
•ar. compl. of fixed points of symmetric morphisms [Frid03]• characterization of un. rec. words of linear ar. compl. [Frid03]• uniformly recurrent words of lowest complexity [Avgustinovich, Cassaigne, Frid, submitted]
• a family with ar. compl. from a wider class (new)• on ar. compl. of Sturmian words (Cassaigne, Frid, preliminary results published)
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WACaM’04 July 17, 2004
Symmetric D0L words
0 011 10
Thue-Morse morphism,ar. compl. of the fixed point is 2n
0 0101 121
3 303
2 232
A symmetric morphism:
ii )0()(
Its fixed point is 010 121 010 121 232 121 ….
of ar. compl. 42 . 2n-2
In general, on the q-letter alphabet aw(n)=q2kn-2, k|q.
A. Frid Growth of arithmetical complexity12
WACaM’04 July 17, 2004
p-adic complexity
is the nimber of words occurring in subsequences of differences of w
Our technique does not work
)(nap kp
Open problem. What is of the Thue-Morse word?
)(3 na
0110 1001 1001 0110 1001 0110…
A. Frid Growth of arithmetical complexity13
WACaM’04 July 17, 2004
? ? ? ? ? ??? ? ? ? ? ? ???
Regular Toeplitz words
P1=ab?cd? a (3-regular) pattern
b ? c d ? baa c d ? a b ?? ?1P
P2=ef? a (3-regular) pattern
b e c d f baa c d e a b f? ?12 PP
A (3-regular) Toeplitz word
,...),(?...lim 2112 PPTPPPnn
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Linearity
Theorem. Let w be a uniformly recurrent infinite word. Then aw(n)=O(n)
iff up to the set of factors w=T(P1,P2,…), where
• all patterns Pi are p-regular for some fixed prime p;• sequence {P1,P2,…} is ultimately periodic
Uniformly recurrent word: all factors occur infinite number of times with bounded gaps
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WACaM’04 July 17, 2004
Another example
P=0?1? 2-regular paperfolding pattern
w=Q·T(P,P,…)=T(P1,P2, P1,P2,…), where
0 3 22 3 2 01w
P1=2?0?3?2?1?3?
Q=23? 3-regular
00 3233 2
P2=3?0?2?3?1?2?
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WACaM’04 July 17, 2004
Lowest complexity?
A word w is Sturmian if its subword complexity is minimal for a non-periodic word: .1)( nnfw
Is arithmetical complexity of a Sturmian word also minimal?
NO, it is not even linear (Sturmian words are not Toeplitz words)
What words have lowest ar. complexity?
A. Frid Growth of arithmetical complexity17
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Relatives of period doubling word
Let a be a symbol, p be a prime integer.Define
and
2w 0100 0101 0100 0100 0100 0101 0100 0100…period doubling word
3w 001001000 001001000 001001001… etc.
Rp(a)=ap-1?
wp=T(Rp(0),Rp(1),…, Rp(0),Rp(1),…)
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Minimal ar. complexity
12262)(22
pp
n
na
pp pw
2 3
p
and these limits are minimal for uniformly recurrent words
[Avgustinovich, Cassaigne, Frid, submitted]
A. Frid Growth of arithmetical complexity19
WACaM’04 July 17, 2004
Plot for ar. complexity of wp
length
ar. c
om
pl.
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Not uniformly recurrent?
All results on linearity are valid only for uniformly recurrent word.
Open problem. Are there (essentially) not uniformly recurrent words of linear arithmetical complexity?
something un. rec. word
is not considered
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More classification
Exponentialar. compl.
Symm. D0L words
Linearar. compl.(un. rec.
characterized)
min
ar. compl.]))([log( nnfO pu
Sturmian words,O(n3)
A. Frid Growth of arithmetical complexity22
WACaM’04 July 17, 2004
Words with aw(n)=O(nfu([logp(n)]))
Recall that for a symbol a and a prime p
Rp(a)=ap-1?.
For u=u0u1…un… let us define
Wp(u)=T(Rp(u0),Rp(u1),…, Rp(un),…).
)(3 uw 000000001 000000001 000000000…
u=0010…
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A theorem
Theorem. For all u (on a finite alphabet) and each prime p>2, aw(u)(n)=O(nfu([logp(n)])).
for p=2, the situation is more complicated since
01010101... may occur both in
and ...)011(2w ...).100(2w
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Particular cases
• If u is periodic, then aw(n)=O(n) which agrees with the characterization above;
• If fu(n)=O(n), then aw(n)=O(n log n) for example, when u is a Sturmian word, or the Thue-Morse word, or 0 1 00 11 0000 1111…;
• If fu(n)=O(n log n), then aw(n)=O(n log n log log n);
• If fu(n)=O(na), then aw(n)=O(n (log n)a);
A. Frid Growth of arithmetical complexity25
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Particular cases - 2
• If fu(n)=O(an), then aw(n)=O(n1+log a); so, on the binary alphabet we can reach aw(n)=O(n1+log 2); for larger alphabets, the degree grows.
• If fu(n) grows intermediately between polynomials and exponentials, then aw(n) grows intermediately between n log n and polynomials.
p
3
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Geometric definition of Sturmian words
is irrational, is arbitrary
All Sturmian words can be constructed so, the set of factors does not depend on c, the subword complexity is n+1
)1,0( )1,0[c
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Subsequence of difference 2
Factors of an arithmetical subsequence also can be represented as intersections of a line with the grid
A. Frid Growth of arithmetical complexity28
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Dual picture: gates and faces
[Berstel, Pocchiola, 93]
A. Frid Growth of arithmetical complexity29
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Counting faces
By Euler formula,
f=e-v+1=
23
)1()1()()(2
1
1
nnnddn
n
d
where is the Euler function)(n
We have )(2/)( 3nOfnaw
A. Frid Growth of arithmetical complexity30
WACaM’04 July 17, 2004
Computational results
It seems that for 1/3< <2/3,
),(2/)( npfnaw
),( npwhere is a simple function, ultimately periodic on .n
For the Fibonacci word
2
15
,...9,8,9,8,9,8,9,8,11,10,8,5,3,1,0,0),( np
A. Frid Growth of arithmetical complexity31
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The current state
Exponentialar. compl.
Symm. D0L words
Linearar. compl.(un. rec.
characterized)
min
ar. compl.]))([log( nnfO pu
Sturmian words,O(n3)
linear subword
complexity
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WACaM’04 July 17, 2004
Other complexities
• d-complexity, Ivanyi, 1987
• pattern complexity, Restivo and Salemi, 2002
• maximal pattern complexity, Kamae and Zamboni, 2002
• modified complexity, Nakashima, Tamura, Yasutomi, 1999
Only complexities which are not less than the subword one: