Additive Energy and Irregularities of Distribution

Abstract

We consider strictly increasing sequences ${\left(a_n\right)}_{n\ge 1}$ of integers and sequences of fractional parts ${\left(\left\{a_n\alpha \right\}\right)}_{n\ge 1}$ where $\alpha \in \mathbb{R}$. We show that a small additive energy of ${\left(a_n\right)}_{n\ge 1}$ implies that for almost all $\alpha$ the sequence ${\left(\left\{a_n\alpha \right\}\right)}_{n\ge 1}$ has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.

8 pages. To appear in Uniform Distribution Theory