Metric results on the discrepancy of sequences (anα)n≥ 1 modulo one for integer sequences (an)n≥1 of polynomial growth

Abstract

An important result of Weyl states that for every sequence (anα)n≥ 1 of distinct positive integers the sequence of fractional parts of (anα)n≥ 1 is uniformly distributed modulo one for almost all . However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy ({αnα}) n≥ 1 of for almost all . In particular, it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences (anα)n≥ 1 and for some special cases such as the Kronecker sequence ({nα}) n≥ 1 or the sequence . In the present paper we answer the question for a large class of sequences (anα)n≥ 1 including as a special case all polynomials an = P(n) with P π ℤ [x] of degree at least 2.