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J OURNAL DE T HÉORIE DES N OMBRES DE B ORDEAUX J. C. L AGARIAS J. O. S HALLIT Linear fractional transformations of continued fractions with bounded partial quotients Journal de Théorie des Nombres de Bordeaux, tome p. 267-279 9, no 2 (1997), <http://www.numdam.org/item?id=JTNB_1997__9_2_267_0> © Université Bordeaux 1, 1997, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 267 Linear Fractional Transformations of Continued Fractions with Bounded Partial Quotients par J.C. LAGARIAS ET J.O. SHALLIT RÉSUMÉ. Soit 03B8 un nombre réel de 03B8 = développement en fraction continue [03B10, 03B11, 03B12,... ], et soit matrice d’entiers tel que 0. Si 03B8 est à quotients partiels quotients partiels bornés. si aj ~ K pour tout j suflisamment grand, alors a*j ~ | det(M)|(K + 2) pour tout j suffisamment grand. Nous donnons aussi une borne plus faible qui est valable pour tout a*j avec j ~ 1. Les demonstrations utilisent la constante d’approximation diophantienne homogène L~(03B8) = une bornés, alors Plus précisément, lim sup q~~ (q~q03B8~)-1. ABSTRACT. Let 03B8 be det M ~ [a*0, a*1, a*2,...] a est aussi à Nous montrons que real number with continued fraction expansion 03B8 = [a0, a1, a2,...], and let be a matrix with integer entries and nonzero determinant. If 03B8 has bounded partial quotients, then [a*0, a*1, a*2, ...] also has bounded partial quotients. More precisely, if aj ~ K for all sufficiently large j, then | det (M)|(K + 2) for all sufficiently large j. We also give a weaker bound valid for all with j ~ 1. The proofs use the homogeneous Diophantine approximation constant L~(03B8) lim sup q~~(q~q03B8~)-1. We show that = det(M)|L~(03B8). Manuscrit re~u le 28 avril 1997 268 1. INTRODUCTION. Let 0 be a real number whose expansion as a simple continued fraction is and set where we adopt the convention that K(0) = +oo if 0 is rational. We say that 0 has bounded partial quotients if K(O) is finite. We also set with the convention that Koo (8) is finite if and K(8), and = +oo if 0 is rational. Certainly Koo (8) ~ only if K(B) is finite. A survey of results about real numbers with bounded partial quotients is given in [17]. The property of having bounded partial quotients is equivalent to 0 being a badly approximable number, which is a number 0 such that in which nearest IlxllI = integer min(x - and q x) through integers. runs This note proves two denotes the distance from x to the quantitative versions of the theorem that if B has b bounded partial quotients and M det(M) # 0, co+d The first result bounds J also has bounded is an integer matrix with partial quotients. in terms of Koo(8) and depends only K ( °’B+6 cO+d) on THEOREM 1.1. Let 0 have an on integer matrix with a bounded partial quotients. 0, then The second result upper bounds the entries of M: K( de+b in terms 0) and P depends 269 THEOREM 1.2. Let 9 have bounded an integer matrix with The last term in ao of 9, since Theorem 1.2 gives det(M) # 0, (1.4) no can partial quotients. If M Chowla [3] proved in 1931 that weaker than Theorem 1.2. partial quotient ao [a d.dJ is Lc then be bounded in terms of the bound for the = partial quotient := 2ad(K(O) +1)3 , I i of a result rather We obtain Theorem 1.1 and Theorem 1.2 from stronger bounds that which appear relate Diophantine approximation constants of 9 and below as Theorem 3.2 and Theorem 4.1, respectively. Theorem 3.2 is a simple consequence of a result of Cusick and Mend6s France [5] concerning the Lagrange constant of 0 (defined in Section 2). can be directly computed from that for The continued fraction of 8, as was observed in 1894 by Hurwitz [9], who gave an explicit formula for the continued fraction of 20 in terms of that of 0. In 1912 Chitelet [2] gave from that of 0, an algorithm for computing the continued fraction of and in 1947 Hall [7] also gave a method. Let M (n, Z) denote the set of n x n E M (2, Z) with Raney [15] gave for each M = a -c d ’ det(M) =1= 0 an explicit finite automaton to compute the additive continued integer b matrices. fraction of cO+d from the additive continued fraction of 0. In connection with the bound of Theorem 1.1, Davenport [6] observed that for each irrational 0 and prime p there exists some integer 0 a p has infinitely many partial quotients an(B’) > p. such that 0’ = 0 Mend6s France [13] then showed that there exists some 0’ = 0 + Pp having the property that a positive proportion of the partial quotients of 0’ have +~ an (0’) 2:: P. Some other related results appear in Mend6s France on continued fractions appear in [1,8,10,18]. [11,12]. Basic facts 2. BADLY APPROXIMABLE NUMBERS Recall that the continued fraction expansion of an irrational real number 270 0 = ~ao, al,...] ] and for n > 1 The n-th is determined by by the recursion complete quotient an The n-th convergent v’ qn of 0 is of 0 is whose denominator is given by the recursion q_ 1 It is well known (see [8, §10.7]) that an+lqn + - Since an+1 + 1 and qn, this 0, qo - 1, and implies that for n > 0. We consider the following Diophantine approximation constants. For irrational number 0 define its type L(O) by and define the grange constant horraogeneous Diophantine approximation of 0 by constant or an La- 271 We use (N.B.: the convention that if B is rational, then some authors study the reciprocal of what L(0) we = L~(B) _ +oo. have called the La- grange constant.) The best approximation properties of continued fraction convergents give The set of values taken by trum [4]. It is well known that L~ (B) then another formula for Loo(6) is see all 0 is called the J5 for all 0. If 0 = over > Lagrange [4,p. 1]. There are simple relations between these quantities and the cf. [16, pp. 22-23]. tient bounds K(O) and LEMMA 2.1. For any irrational 0 with bounded Proof. This is immediate from (2.2) and (2.3). LEMMA 2.2. For any irrational 0 with bounded Proof. This is immediate from do not Although we in spec- [ao, at, a2, ... ~, (2.7) can be use (2.2) it in the slightly improved. and (2.4). sequel, Since partial quo- partial quotients, we have 0 partial quotients 0 note that both inequalities (an -I- l)qn-l, (2.1) yields we 272 Since from Next, from (2.1) we some point (2.4) yield on, this and have Hence Let K = K,,. (0). Then for all n sufficiently large we have so We conclude that 3. LAGRANGE CONSTANTS An integer matrix M fractional transformation Note that Ml(M2(8)) = = La on AND [c doj PROOF THEOREM 1.1. I with det(M) # 0, a real number 8 MlM2(8). OF by acts as a linear 273 LEMMA 3.1. If M is an integers matrix with det(M) grange constants of 0 and M(O) are related by This is D Proof. (2.5). = ~1, then the La- well-known, cf. [14] and [5, Lemma 1], and is deducible from The main result of Cusick and Mend6s France [5] yields: THEOREM 3.2. For any integers m > 1, let Then for any irrational number 0, and Proof. Theorem 1 of [5] states that Let GL(2, Z) denote the group of 2 x 2 integer matrices with determinant ~1. We need only observe that for any M in Gm there exists some M E GL(2, Z) so, such that 1Vl M and 0 = whence = 0 d’ 0U d then Lemma 3.1 (3.4) implies (3.2). with a‘d’ = m and 0 b’ d’ . For if gives To construct M =LC we must have 274 Take C = and complete this row matrix0 1 c ±l ’ Then gcd(C, D) - 1, so we may matrix M E GL(2, Z). Multiplying this by a suitable the desired M. y yields to The lower bound adjoint matrix a (3.3) follows from the upper bound - We prove by contradiction. M E Gm and some 0 we have Suppose (3.3) (3.2). We use the - were false, so that for some This states that which contradicts (3.2) for 0’, since det(M’) = det(M) = 0 m. Remark. The lower bound (3.3) holds with equality for some values of 0 and not for other values. If for given 0 we choose an M E Gm which gives = then equality holds in (3.3) equality in (3.2), so that = for B’ adj(M)(0). However, if as then V5, occurs for 0 for all M; hence (3.3) does not hold with equality when m > 2. = = Proof of Theorem 1.1. Theorem 3.2 gives apply Lemma 2.2 twice to get To obtain the lower and apply (3.5) :5 bound, we use the adjoint with M’ and 0’ = M(O) J to obtain Now 275 Since this I det (M) I = yields 4. NUMBERS OF BOUNDED TYPE Recall that the type L(8) integer matrix with Proof. Set 1b = Then L(0) > 1, PROOF OF THEOREM 1.2 of 0 is the smallest real number such that THEOREM 4.1. Let 0 have bounded an AND det(M) # 0, partial quotients. then Suppose first that c = 0 so that I det(M)1 = ladl where We have For any e > 0 we (4.2) may choose q in Letting c -~ 0 yields (4.1) when c 0. Again Suppose now that = so that ~ ~ L(1f;) - 0. L(~) > ~ where e. Then > 0. 276 We have so that We first treat the case = 0. Now det(M) # lPd .J (4.5) since det hence qa - pc 0. Thus if qa - pc = = 0 then 1, gives It follows that qa - pc ,-~ 0 provided that ~/ We next treat the case when qa - pc 54 0. Now from the definition of we see Given (4.6) e > and However, 0, we (4.9) may that the bound choose q so that ~ ~ L(~) - e, and we obtain from 277 implies that Multiplying this by §q Letting e -7 provided that The last two Proof of 0 and (4.8) and applying it holds. Now directly (4.8) fails to hold when Applying Theorem 4.1 which is the desired bound. 0. only if 0 and Lemma 2.1 (1). gives 0 The proof method of Theorem 4.1 the bounds prove Remarks. (4.10) yields using q > 1 yields inequalities imply (4.1) Theorem to the left side of can also be used to of Theorem 3.2, from which Theorem 1.1 can be easily deduced. The lower bound in (4.14) follows from the upper bound as in the proof of Theorem 3.2. We sketch a proof of the upper bound in (4.14) for the case 278 sufficiently large q* > q* (E* ), have we We choose q as n -~ (2). For = qn (1b) for sufficiently large n, and note that is irrational. We can then replace (4.9) by (4.15), and +,E*. Letting q -+ oo, (4.11) with L(O) replaced by -~ 0 in that order yields the upper bound in (4.14). since 0 oo, then deduce f -+ 0 and E* By 0. For any c* > 0 and all with 9 = a given matrix M consider the set of attainable ratios Lemma 3.1 the set V(M) depends only on its SL(2, Z)-double coset Theorem 3.2 shows that It is and an Both I follows from Theorem 3.2 and the remark interesting open problem to determine the set I lie in V(M), as following it. Acknowledgment. We are indebted to the referee for helpful comments and references, and in particular for raising the open problem about V(M) . REFERENCES 1. A. 2. Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, 1984. A. Chitelet, Contribution à la théorie des fractions continues arithmétiques, Bull. Soc. Math. France 40 3. S. D. Chowla, Some 33 (1931), 544-563. (1912), 1-25. problems of diophantine approximation (I), Math. Zeitschrift 279 4. T. W. Cusick and M. Flahive, The Markoff and Lagrange Spectra, American MathSociety, Providence, RI, 1989. Cusick and M. Mendès France, The Lagrange spectrum of a set, Acta Arith. ematical 5. T. W. 34 (1979), 287-293. 6. H. Davenport, A remark 344. 7. M. Hall, On the 966-993. sum on continued fractions, and product Michigan Math. J. 11 (1964), of continued fractions, Annals of Math. 48 343- (1947), Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford University Press. A. 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Shallit, Continued fractions with bounded partial quotients: a survey, Enseign. Math. 38 18. H. M. (1992), 151-187. Stark, Introduction to Number Theory, Markham, 1970. J. C. Lagarias AT &T Labs - Research, Room C235 180 Park Avenue, P. (J. Box 971 Florham Park, NJ 07932-0971, USA email: [email protected] J. 0. Shallit Department of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada email: [email protected]