Arithmetic properties of automata: regular sequences.
Arithmetic properties of certain power series with algebraic coefficients
Arithmetic properties of positive integers with fixed digit sum.
In this paper, we look at various arithmetic properties of the set of those positive integers n whose sum of digits in a fixed base b > 1 is a fixed positive integer s. For example, we prove that such integers can have many prime factors, that they are not very smooth, and that most such integers have a large prime factor dividing the value of their Euler φ function.
Arithmetic properties of the solutions of a class of functional equations.
Arithmetical investigations of a certain infinite product
Arithmetical properties of gap series with algebraic coefficients
Arithmetical properties of the values of functions satisfying certain functional equations of Poincaré
Arithmetische Eigenschaften ganzer Funktionen mehrerer Variablen.
Arithmetische Eigenschaften spezieller Heinescher Reihen.
Around the Littlewood conjecture in Diophantine approximation
The Littlewood conjecture in Diophantine approximation claims thatholds for all real numbers and , where denotes the distance to the nearest integer. Its -adic analogue, formulated by de Mathan and Teulié in 2004, asserts thatholds for every real number and every prime number , where denotes the -adic absolute value normalized by . We survey the known results on these conjectures and highlight recent developments.
Asymptotic behavior of the number of solutions for non-Archimedean Diophantine approximations
Asymptotic growth of Markoff-Hurwitz numbers
Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values
We present asymptotic representations for certain reciprocal sums of Fibonacci numbers and of Lucas numbers as a parameter tends to a critical value. As limiting cases of our results, we obtain Euler’s formulas for values of zeta functions.
Asymptotisches Verhalten einer diophantischen Approximations-Funktion
Automates et valeurs de transcendance du logarithme de Carlitz
Automates finis et ensembles normaux
Soit une suite strictement croissante d’entiers reconnaissable par un automate fini. Nous montrons qu’une condition nécessaire et suffisante pour que l’ensemble normal associé a soit exactement est que l’un au moins des sommets qui reconnaît la suite soit précédé dans le graphe de l’automate par un sommet possédant au moins deux circuits fermés distincts. Cette condition peut se traduire quantitativement en disant que la suite doit être plus “dense” que toute suite exponentielle.
Automatic continued fractions are transcendental or quadratic
We establish new combinatorial transcendence criteria for continued fraction expansions. Let be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients of is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.
Autour de la conjecture de Zilber-Pink
Nous dressons un rapide panorama de résultats allant dans le sens de la conjecture suivante : l’intersection d’une sous-variété d’une variété semi-abélienne et de l’union de tous les sous-groupes algébriques de de codimension au moins n’est pas Zariski-dense dans dès que n’est pas contenue dans un sous-groupe algébrique strict de .
Averages of short exponential sums