ON THE RATIONAL APPROXIMATION TO THUE–MORSE RATIONAL NUMBERS

. Let b ≥ 2 and (cid:96) ≥ 1 be integers. We establish that there is an absolute real number K such that all the partial quotients of the rational number (cid:96) (cid:89) h =0 (1 − b − 2 h ) , of denominator b 2 (cid:96) +1 − 1 , do not exceed exp( K (log b ) 2 √ (cid:96) 2 (cid:96)/ 2 ).


Introduction
An easy covering argument which goes back to Cantelli shows that, for almost all real numbers ξ (with respect to the Lebesgue measure) and for every positive ε, the inequality ξ − p q > 1 q 2+ε holds for every sufficiently large q.However, it is often a very difficult problem to show that a given real number shares this property, unless its continued fraction expansion is explicitly determined.This is known to be the case for any irrational real algebraic number, by Roth's theorem, and for only a few other real numbers defined by their expansion in some integer base.Let t = t 0 t 1 t 2 . . .denote the Thue-Morse word over {−1, 1} defined by t 0 = 1, t 2k = t k and t 2k+1 = −t k for k ≥ 0.Then, the Thue-Morse generating series ξ t (z) is given by By means of a non-vanishing result obtained in [1] for the Hankel determinants associated with the Thue-Morse sequence, Bugeaud [6] established that, for any given positive ε and any integer b ≥ 2, the Thue-Morse-Mahler number , for every sufficiently large q.Subsequently, his result has been considerably improved by Badziahin and Zorin [4], who showed that there exists a positive real number K such that the stronger inequality holds as soon as q is large enough.Thus, all the partial quotients of ξ t (b) are rather small.Note that, in view of [4,Th. 11], the number K occurring in [4, Th. 2] must depend on b and can be taken equal to log b times some number depending only on the series f (z) occurring in the statement of [4,Th. 2].
Observe that the Thue-Morse power series ξ t (z) is the limit of the sequence of rational functions More precisely, we have and Let b ≥ 2 and ≥ 1 be integers.For a rational number p/q, we derive from (1.2) that Consequently, ξ t (b) and f (b) have the same first partial quotients.To see this, let p n /q n be the convergent to ξ t (b) with q n ≤ b 2 and n maximal for this property.We assume that is sufficiently large to ensure that n ≥ 8.A short calculation shows that (1.4) q n ≥ q n−1 + q n−2 ≥ . . .≥ 8q n−5 .
By a result of Borel [12, Ch.I, Th. 5B], there exists ε in {0, 1, 2} such that It then follows from (1.3) and (1.4) that which, by a classical theorem of Legendre [12, Ch.I, Th. 5C], implies that p n−5−ε /q n−5−ε is a convergent of f (b).Consequently, ξ t (b) and f (b) have the same n − 7 first partial quotients.By (1.1), these partial quotients are rather small.However, (1.1) gives no information on the remaining partial quotients of f (b), thus, in particular, on the rate with which the rational number f (b) of denominator b 2 +1 −1 is approximated by rational numbers p/q of denominator q greater than b 2 .In the present note, we address this question and show that an inequality like (1.1) remains true for every convergent of f (b).
Theorem 1.1.There exists a positive real number K such that, for every integer b ≥ 2 and every integer ≥ 2, the inequality h=0 holds for every rational number p/q different from f (b).Write with a ( ) The partial quotients a ( ) . There exists a positive real number C, depending only on b, such that the length L( ) of the continued fraction of f (b) exceeds The second assertion of Theorem 1.1 immediately follows from the first one and the theory of continued fractions.The last assertion already follows from (1.1).
Theorem 1.1 is mainly motivated by the very few known results on continued fraction expansions of sequences of rational numbers.Pourchet [10] (see also [5,11]) proved that, for all coprime integers a and b with 1 < b < a and for every positive ε, there exists a positive C, depending only on ε and on the prime divisors of a and b, such that all the partial quotients of (a/b) n are less than Cb εn .This was subsequently extended to quotients of power sums by Corvaja and Zannier [7], with a similar conclusion.Consequently, the length of the continued fraction expansion of (a/b) n (resp., of The function q → exp( √ log q log log q) occurring in (1.1) is a consequence of the bound of order (c 1 k) c 2 k obtained in [4] for the absolute values of the coefficients of the numerator and denominator of the k-th convergent to ξ t (z).Numerical experiments suggest that a better bound of the shape c √ k 3 should hold (here, c 1 , c 2 and c 3 are absolute, positive real numbers).Such a result seems difficult to establish.See Figure 1.
To prove (1.1), Badziahin and Zorin [4] used that all the partial quotients of the continued fraction expansion of ξ t (z) are linear, a result established by Badziahin [2].Here, we first show that all the partial quotients of the rational functions f (z), ≥ 1, are linear.This is the main novelty of the present note and the object of Section 2.Then, in Section 3, we establish Theorem 1.1 by adapting to our purpose the argumentation of [4].Finally, in the last section, we discuss another example.
be the family of rational numbers defined in Proposition 2.1.Then, For ≥ 0, all the partial quotients in the continued fraction expansion of g (z) are polynomials of degree one.
The last assertion of Theorem 2.2 immediately follows from Proposition 2.1.
Proof.We prove identity (2.1) by induction on .Since ) is true for = 0, 1.Let k ≥ 1 be an integer and suppose that (2.1) is true for It suffices prove that g k+1 (z) = h k+1 (z).From the even contraction theorem (see, for example, [9, Theorem 2.1(1)]), we have where (z − 1), By removing the common factors in numerators and denominators, we obtain where , Using the recurrence relations defined in the statement of Theorem 2.2, a quick calculation shows that we have This implies that h k+1 (z) = (z − 1)g k (z 2 ) = g k+1 (z).The key new ingredient for the proof of Theorem 1.1 is the fact that all the partial quotients of the rational functions f (z) are linear.This allows us to follow the argumentation of [4], with some minor changes.For the sake of readability, we keep most of the notation of [4] and we sketch how to adapt the proof of [4,Th. 11].Instead of working with the (infinite) power series g u (z), we fix a positive integer and work with the (finite) power series Since, by Theorem 2.2, all the partial quotients of g (z) are linear, the auxiliary results in [4] hold.Furthermore, for m ≥ 1, we have Denoting by p k, (z)/q k, (z) the convergents to q (z), where k = 1, . . ., 2 +1 , and defining p k, ,m (z) and q k, ,m (z) as in [4, (2.20)], the analogue of [4, (2.25)] holds, namely, we have . By [4, Lemma 9], the integer q k, ,m is controlled and is comparable to b k2 m .Take now a large integer L (which corresponds to the integer in the statement of the theorem) and study the rate of approximation to the rational number g L (b) of denominator b 2 L+1 by rational numbers p/q of denominator less than b 2 L+1 .We follow the argument of [4] and look for a power of b close to q.We use the fact that every integer n less than 2 L+1 is rather close to a product k2 m , where L = m + and k ≤ 2 +1 .The latter constraint comes from the construction of our finite sequence of good rational approximations to g L (b).It is not required to hold in [4].
Let p/q be a rational number with q < b 2 L+1 and q sufficiently large (it is sufficient to assume that q exceeds b κ 1 , for some absolute constant κ 1 ) to guarantee that the real number x defined in [4, (3.2)] satisfies x > b 2 .
This inequality holds if for a suitable positive κ 2 , depending only on τ , we have (3.2) q < exp −κ 2 log b log 2 q log 2 log 2 q b 2 L+1 .
There exist integers n and m and a real number α such that 2 m divides n and t ≤ n ≤ t + 2τ t log 2 t, 2 m = ατ t log 2 t, 1 ≤ α ≤ 2.

Figure 1 .
Figure 1.Logarithm of the absolute values of the coefficients of the k-th convergent