EQUIDISTRIBUTION MODULO 1 AND SALEM NUMBERS

Christophe Doche (1) , Michel Mendes-France (2) , Jean-Jacques Ruch (2)

1 ACAC - Centre for Advanced Computing - Algorithms and Cryptography
2 IMB - Institut de Mathématiques de Bordeaux

Let θ be a Salem number. It is well-known that the sequence (θ n ) modulo 1 is dense but not equidistributed. In this article we discuss equidistributed subsequences. Our first approach is computational and consists in estimating the supremum of limn→∞ n/s(n) over all equidistributed subsequences (θ s(n) ). As a result, we obtain an explicit upper bound onthe density of any equidistributed subsequence. Our second approach is probabilistic. Defining a measure on the family of increasing integer sequences, we show that relatively to that measure, almost no subsequence is equiditributed.

Mots clés: Salem number; Equidistribution modulo 1; J0 Bessel function.; J0 Bessel function

Domaines: Théorie des nombres [math.NT]

Christophe Doche, Michel Mendes-France, Jean-Jacques Ruch. EQUIDISTRIBUTION MODULO 1 AND SALEM NUMBERS. Functiones et Approximatio, 2008, XXXIX (2), pp.261-271. ⟨hal-00353834⟩