Discrepancy estimates for some linear generalized monomials

Abstract

We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also contain the integer part operation [·] additionally to the addition and the multiplication. A well-studied example is the (nα) sequence defined by the monomial αx. Their most basic sister — ([nα]β)n≥0 — is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper. We show in particular that if the pair of real numbers (α, β) is in a certain sense badly approximable, then the discrepancy satisfies a bound of order Oα,β,e(N).