On approximation measures of q-exponential function

We shall present effective approximation measures for certain infinite products related to [Formula: see text]-exponential function [Formula: see text]. There are two main targets. First we shall prove an explicit irrationality measure result for the values [Formula: see text], where [Formula: see text], and [Formula: see text], [Formula: see text]. Then, if we restrict the approximations of [Formula: see text] to rational numbers of the shape [Formula: see text], where [Formula: see text], we may replace Bundschuh’s irrationality exponent [Formula: see text] by [Formula: see text]

On rational approximations of certain Mahler functions with a connection to the Thue–Morse sequence

We shall obtain the irrationality exponent 2 for some values of two special Mahler functions. This gives a new proof for the recent result of Bugeaud on Thue–Morse–Mahler numbers.

The Euclid Algorithm is totally gaussian

International audience We consider Euclid’s gcd algorithm for two integers $(p, q)$ with $1 \leq p \leq q \leq N$, with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function $d$ on the set of possible digits, asymptotically for $N \to \infty$. For any additive cost of moderate growth $d$, Baladi and Vallée obtained a central limit theorem, and –in the case when the cost $d$ is lattice– a local limit theorem. In both cases, the optimal speed was attained. When the cost is non lattice, the problem was later considered by Baladi and Hachemi, who obtained a local limit theorem under an intertwined diophantine condition which involves the cost $d$ together with the “canonical” cost $c$ of the underlying dynamical system. The speed depends on the irrationality exponent that intervenes in the diophantine condition. We show here how to replace this diophantine condition by another diophantine condition, much more natural, which already intervenes in simpler problems of the same vein, and only involves the cost $d$. This “replacement” is made possible by using the additivity of cost $d$, together with a strong property satisfied by the Euclidean Dynamical System, which states that the cost $c$ is both “strongly” non additive and diophantine in a precise sense. We thus obtain a local limit theorem, whose speed is related to the irrationality exponent which intervenes in the new diophantine condition. We mainly use the previous proof of Baladi and Hachemi, and “just” explain how their diophantine condition may be replaced by our condition. Our result also provides a precise comparison between the rational trajectories of the Euclid dynamical system and the real trajectories.

Irrationality measures for some automatic real numbers

This paper is devoted to the rational approximation of automatic real numbers, that is, real numbers whose expansion in an integer base can be generated by a finite automaton. We derive upper bounds for the irrationality exponent of famous automatic real numbers associated with the Thue–Morse, Rudin–Shapiro, paperfolding and Baum–Sweet sequences. These upper bounds arise from the construction of some explicit Padé or Padé type approximants for the generating functions of these sequences. In particular, we prove that the Thue–Morse–Mahler numbers have an irrationality exponent at most equal to 4. We also obtain an explicit description of infinitely many convergents to these numbers.