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Abstract. This paper studies tilings related to the β-transformation when β is a Pisot number (that is not supposed to be a unit). Then it applies the obtained results to study the set of rational numbers having a purely periodic β-expansion. Secial focus is given to some quadratic examples.

We extend previous results of Delange (Acta Arith. 21, 1972, 285 298) concerning the existence of distribution functions of certain q-adic digital functions to the case of digital expansions with respect to linear recurrences with decreasing coefficients. Furthermore we investigate a special case of digital functions and give a functional equation for the related distribution function. We prove uniqueness and continuity of the solution of this equation.

Dedicated to the memory of Pierre Liardet, our friend and teacher, who introduced us to the subject.International audienceLet G = (G(n))(n) be a strictly increasing sequence of positive integers with G(0) = 1. We study the system of numeration defined by this sequence by looking at the corresponding compactification K-G of N and the extension of the addition-by-one map tau on K-G (the 'odometer'). We give sufficient conditions for the existence and uniqueness of tau-invariant measures on K-G in terms of combinatorial properties of G

The distribution of binomial coefficients in residue classes modulo prime powers and with respect to the p-adic valuation is studied. For this purpose, general asymptotic results for arithmetic functions depending on blocks of digits with respect to q-ary expansions are established.