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 (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

Gaussian behavior of quadratic irrationals

“…But we also present in Section 5 similar results that hold for rqi numbers with bounded digits. These results combine the general methods of this paper with previous results that are specific to the constrained case, described in [11,27,4]. In the same section, we also study an important (non-additive) cost, namely the Lévy constant, and obtain asymptotic expansions for its expectation and its variance.…”

“…Then, the present authors [4] studied the probability π(N, M ) in the rational case. Besides some specifications and extensions of a first result obtained by Hensley [10], they developed a methodology (based on principles of analytic combinatorics) which can be easily transferred from the rational case to the rqi case and implies the following general result: This result exhibits a threshold phenomenon, already obtained by Hensley and precisely described in [4] in the rational case, depending on the relative order of σ M − 1 (of order O(1/M )) with respect to n := log N : (a) If M/n → ∞, then almost any number of size at most N has all its digits less than M . (b) If M/n → 0, then almost any number of size at most N has at least one of its digits greater than M .…”

“…Moreover, if M is large enough (say M ≥ M 0 ), the real number σ M is close enough to 1, and the vertical line s = σ M is contained in the Dolgopyat strip of Theorem A. Then, bounds à la Dolgopyat are available for the constrained transfer operator (which can be viewed as a perturbation of the unconstrained one, as in [4]). This entails the polynomial growth of the quasi-inverse (I − H M,t,w ) −1 , and thus the polynomial growth of Q M (s, w), P M (s, w) for | s| → ∞ and w close to 0.…”

Combinatorics, Words and Symbolic Dynamics

“…has been precisely studied in the paper (Cesaratto and Vallée, 2011) for a general function M = M(N).…”

Gaussian Behavior of Quadratic Irrationals

“…This is not completely exact, as the final operator Fs,w also intervenes (see Section 4.4) 4. The paper[1] does not use the "ready for use" version of the Landau Theorem described in Section 4.2, and, as in many other works (see for instance[21]), it proves the analog result "by hands".…”