Small quotients in Euclidean algorithms

International audienceNumbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension σ M of reals with digits in their continued fraction expansion bounded by M was considered, and estimates of σ M for M → ∞ were provided by Hensley [12]. In the rational case, first studies by Cusick, Hensley and Vallée [4, 9, 19] considered the case of a fixed bound M when the denominator N tends to ∞. Later, Hensley [11] 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 ex-pansion) 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 re-striction 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 generalizes previous estimates due to Hensley [12] about the Haus-dorff dimension σ M to the case of other continued fraction expansions. The method used in the paper combines technics from analytic combinatorics and dynamical systems and it is an instance of the Dynamical Analysis paradigm introduced by Vallée [20], and refined by Baladi and Vallée [2]