A note on trace of powers of algebraic numbers

Abstract

We study pairs of non-zero algebraic numbers (α,λ) such that TrQ(α,λ)/Q(λαn)∈Z for n lying in suitable subsets I of N. In this short note, we give a satisfactory account to these circle of questions. This theme arises naturally in the study of distribution of exponential sequences modulo one. Finally we indicate that the analogous results do not hold for global fields in positive characteristic.