Almost-arithmetic progressions and uniform distribution

Abstract:

In a recent paper, P. E. O’Neil gave a new criterion for uniform distribution modulo one in terms of almost-arithmetic progressions. We investigate the relation between almost-arithmetic progressions and uniformly distributed sequences from a quantitative point of view. An upper bound for the discrepancy of almost-arithmetic progressions is given which is shown to be best possible. Estimates for more general sequences are also obtained. As an application, we prove a quantitative form of Fejér’s theorem on the uniform distributivity of slowly increasing sequences.