Arch. Math.. Vol. 40, 319--323 ( 1 9 8 3 ) 0003-889X/83/4004-0319 $ 2.50/0 �9 1983 Birkhiiuser Verlag, Basel

A characterization of the semiregular continued fractions which represent quadratic irrationals

By

ULRICH HOLZBAUR and Max RIEDERLE

Introduction. Throughout this paper, ~ denotes an irrational number which is represented by the regular continued fraction [b0*, hi*, ...]. Moreover, if (Xn) is any sequence, we write H (Xn) for the set of all its real limit points.

I t is well-known that the convergents A*n/B*~ to ~ satisfy the condition I ~ -- A*n/B*~ I 1/(B~*) 2. Hence, for a more detailed study of their approximation properties it is

~,~.= Bn (Bn ~ -- A'n). An natural to investigate the limit points of the sequence *" * * important result in this context is due to Jurkat and Peyerimhoff [3]:

The number ~ is a quadratic irrational if and only if H (6*n) is finite and does not contain zero.

In this paper, we will ~ve a generalization of this result to milnite semiregmlar

continued fractions, i.e. : continued fractions of the form b0 + -.. with the properties:

(SR1) a~e{4-1} for v e ~ ,

(Sg2) b0eT/, b ~ N , b ~ + a ~ + l ~ l for v e ~ ,

(SR 3) b~ + a~+i ~ 2 for infinitely many v.

According to a Theorem of Tietze [9] every infinite semiregular continued fraction is convergent to some irrational number. Conversely, if ~ is an arbitrary irrational number and (av) is any prescribed sequence of units there exists exactly one semi-

regular continued fraction with: ~ = b0 + . . . . Furthermore, our considera-

tions are valid for the continued fractions to nearest and to farthest integers, the diagonal continued fractions and the singular continued fractions.

Throughout this paper, we assume now that b0 ~ ~b~ ~ "'" is any fixed semi- I -

regular continued fraction representing ~. We denote its convergents by An/Bn and write 8n -~ Bn(Bn~ -- An). With these notations, our result reads as follows:

Theorem. The number ~ is a quadratic irrational i / a n d only i / H (~n) is finite and does not contain zero.

320 U. HOLZBAUR and M. RIEDERLE ARCH. MATH.

Obviously, this Theorem characterizes the semiregular continued fractions which represent quadratic irrationals although they are, in general, not periodic. The main tool of the proof will be the study of the connections between the sequences ($~)

a~+ll I I and (~*) where ~ = b~ + -b~+~ + " " and ~* - - b* + ~ + - . - . More precisely,

we apply the following Theorem of Blumer [2]:

For every ~ ~ No, we have:

and

~,~{~* *, ~*-~+1 , k = l , 2 . . . . . b* - - l ; x e No} .

The proof of the sufficient part. In this section, we assume tha t H(6~) is finite and does not contain zero and deduce from this tha t $ is a quadratic irrational. To this end, we make use of the following well-known formulas which are valid for all n e N :

(1) ~n > 1.

an+l (2) ~n = b~ + ~n+~

Bn (3) On . - B~-I > O,

(4) 0n----(--1) n a l ' . . . ' a n + l #n+l + a.+l/en '

al � 9 �9 �9 an

(5) On + an+l/$n+l " 6n-i ---- (-- I) n-1

Our first result is

Lemma 1. Suppose that H (CSn) i s / in i te . Then (On) is bounded.

P r o o f . We assume the contrary and distinguish two cases.

(i) Let lira 15n I = ~ " Then, according to (1)--(4), there must exist some number no such tha t an = - - 1 for all n ~ no. Therefore, it follows from (SR 2) that

Bn - - B n - 1 = (bn -- 1)Bn-1 - - Bn-2 >--__ Bn-1 -- Bn-2

for all n --> no. The sequence (Bn -- Bn-1) even tends to infinity since (SR 3) guarantees the existence of a sequence (ni) with bn, _>_ 3. Thus we have Bn > Bn-1 for all sufficiently large n which is equivalent to ~n > 1. Therefore, we may conclude from (2) and (4) tha t

1 I 0., I < ~ , + 1 - 1/o., <= 1

for all sufficiently large i in contradiction to lim Jdn]-----oo.

Vol. 40, 1983 Semiregular continued fractions and quadratic irrationals 321

(ii) Suppose now that H(an) is nonvoid. This guarantees the existence of a sequence (n~) such that [ ~n, ] -> r162 and ~n,-1 -- 0 (1). Without loss of generality we may assume that an,+1 ---- -- 1 for all i e N0. Therefore, we may conclude from (4) that

] i /~ . , ] = ~.,+1 - 1/Q., ~ 0

which implies that (~n,) is bounded and

~,,,+i ~ , - - 1 i ~ 0 .

But then we obtain from (1) and (5) that

R e m a r k s . 1. Actually, the hypothesis "H(~n) is finite" was not needed but only the weaker assumption "H (6n) is bounded".

2. Lemma 1 implies, in particular, tha t H (an) must be nonvoid. Therefore, (an) always contains a bounded subsequence.

Our next result provides a connection between the sequences (~n) and (an).

Lemma 2. Suppose that H (an) is /inite and does not contain zero. Then (~n) is bounded and H (~n) is/inite.

Proof . The combination of the formulas (4) and (5) shows that

( ~n+l ~ n - - a n + l )2 (6) 1 ~- 4 an a,-1 = \ ~n+l Qn + a,+l

for all n ~ ~, hence

{ 1 • + 4anan-1 / (7) ~n+l~ ( - 1)hal "--- "an+l 2a ,

for all n ~ N. The assertion of Lemma 2 now easily follows from this and Lemma 1 by continuity arguments.

R e m a r k . Note that the sign in (7) is not uniquely determined. In fact, according to (6), it is plus ff and only ff I ~n+l Qn I > 1. Therefore, simple examples show that both signs may occur.

Lemma 3. Suppose that (~n) is bounded and H (~n) is ]inite. Then ~ is a quadratic irrational.

Proof . I t was shown by Ballieu [1] (compare also [4] and [8]), that ~ is a quadratic irrational if and only if the sequence (~*) is bounded and H (~n*) is finite. According to Blumer's Theorem, these conditions are fulfilled under our hypotheses unless 1 E H(~n). In order to show tha t this cannot occur, we assume the contrary and distinguish two cases:

Archly der Mathematik 40 21

322 U. HOLZBAUR and M. RIEDERLE "ARCH. MATH.

(i) I f H(~n) contains more t h a n one element, we m a y find a n u m b e r ~. ~ 1 and a sequence (n~) such t h a t ~ , - ~ 1 and ~n,+l--> ~ as i--> co. For sufficiently large i, we have ~n, e (1, 2). Therefore, according to (1) and (2), we have

~ n , = l + l / ~ n , + l or ~ n , = 2 - - 1 / ~ n ~ + t

for those i. Only the second possibil i ty can occur infinitely of ten since (~n) is bounded. But this would lead to ~ = 1.

(ii) I f H(~n) = {1}, we have lira ~n = 1 ; hence, as before, ~n = 2 - - 1/~n+l for all sufficiently large n. Bu t t hen a contradict ion arises f rom

Obviously, the sufficient p a r t of our Theorem now follows a t once f rom the com- binat ion of the L e m m a s 1 to 3.

The proof of the necessary part . Assume now t h a t ~ is a quadra t ic irrational. Because of (4) the calculation of H (cSn) m a y be carried out b y means of the sequences (~n) and (an+l/o~n). These sequences are connected by the following formula of Per- ron [7]: Le t ~ be a quadrat ic i r ra t ional and let ~n+l denote the algebraic conjugate of ~n+l. Then

(8) an+l/~n ---- - - ~n+l (1 + o (1))

as n --> oo. Therefore, it suffices to show t h a t (~n) takes only finite n u m b e r of values. Bu t this follows immedia te ly f rom the Theorems of Blumer and Lagrange.

R e m a r k s . 1. Note t h a t (4) and (8) allow the explicite calculation of H(] ~n])- In fact, i f H(~)----- (x~ [~" = 1 . . . . . r} t hen H( I ~nl)---- { 1 / ( x ~ - - ~ ) I?'---- 1, . . . , r } .

2. An other approach to our Theorem could be obta ined b y combining results of BaUieu [1], Lekkerkerker [4], Negoescu [5] and Tietze [9].

Reterences

[1] R. Baz~LI~U, Sur des suites des nombres li~es ~ une fraction continue r~guli~re. Acad. Roy. Belg. Bull. C1. Sci. (5) 29, 165--174 (1943).

[2] F. BI, V~Fa~, L?ber die verschiedenen Kettenbruchentwicldungen beliebiger reeller Zahlen und die periodischen Kettenbruchcntwicklungen quadratischer Irrationalitgten. Acta Arithmetica g, 3--63 (1938).

[3] W. B. Jv~x~T and A. PEYm~I~HOFF, Characteristic approximation properties of quadratic irrationals. Int. J. Math. and math. Sci. 1,479--496 (1978).

[4] C. G. L~XXv.I~K~mXE1L Una questione di approssiraazione diofantea e una proprieta carat- teristic~ dei numeri quadratici I, II . Atti Acead. Naz. Lincei. Rend. C1. Sci. :Fis. Mat. Hat. (8) 21, 179--185, 257--262 (1956).

[5] :N. :N~GOESCU, Generalization of Theorems of Dirichlet, Vahlen and Hurwitz-Borel on the approximation of irrationals by rationals. Acad. 1%. P. Romine Fil. Iasi Stud. Cerc. Sti. Mat. 13, 219--228 (1962).

[6] O. P~m~o~, Die Lehre yon den Kettenbriichem Stuttgart 1954, 1957. [7] O. Pm~ROI% Neue Periodiziti/tsbeweise fiir die regelm~Bigen und ha].bregelmiil3igen Ketten-

brfiche quadratischer Irrationalzahlen. Sb. der Bayer. Akad. Wiss. Miinchen math. phys. Klasse, 321--333 (1954).

Vol. 40, 1983 Semiregular continued fractions and quadratic irrationals 323

[8] M. I~IEDERLE, Short proofs of theorems of Lekkerkerker and Ballieu. Int. J. of Math. and math. Sei. ~, (3), 609--612 (1982).

[9] H. TIETZE, ~ber Kriterien ffir Konvergenz und Irrationalit~t unendlieher Kettenbriiehe. Math. Ann. 70, 236--265 (1911).

Ansel~ift der Autoren:

Ulrich Holzbaur Reschweg 1 D-7900 Ulm-Lehr

Eingegangen am4.11.1980

Max Riederle EberhardtstraBe 14/3 D-7900 Ulm

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