Small quotients in Euclidean algorithms

Abstract

Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension σ M of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of σ M for M→∞ were provided by Hensley (J. Number Theory 40:336–358, 1992). In the rational case, first studies by Cusick (Mathematika 24:166–172, 1997), Hensley (In: Proc. Int. Conference on Number Theory, Quebec, pp. 371–385, 1987) and Vallée (J. Number Theory 72:183–235, 1998) considered the case of a fixed bound M when the denominator N tends to ∞. Later, Hensley (Pac. J. Math. 151(2):237–255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions; second, it deals with various probability settings (and not only the uniform probability); third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M 0; fourth, it refines the estimates due to Hensley, in the cases that are studied by Hensley. This paper also generalises previous estimates due to Hensley (J. Number Theory 40:336–358, 1992) about the Hausdorff dimension σ M to the case of other continued fraction expansions. The method used in the paper combines techniques from analytic combinatorics and dynamical systems and it is an instance of the Dynamical Analysis paradigm introduced by Vallée (J. Théor. Nr. Bordx. 12:531–570, 2000), and refined by Baladi and Vallée (J. Number Theory 110:331–386, 2005).