Additive Energy and Irregularities of Distribution

TL;DR

This paper explores how the additive energy of integer sequences influences the distribution irregularities of their fractional parts when multiplied by real numbers, revealing that low additive energy leads to large discrepancy for most real numbers.

Contribution

It establishes a link between small additive energy of sequences and large discrepancy in fractional parts for almost all real numbers, with general results and counterexamples.

Findings

Small additive energy implies large discrepancy for almost all α

Provides examples illustrating the relationship

Shows the converse does not always hold

Abstract

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