Additive Energy and Irregularities of Distribution

2016/08/24 by Aistleitner, Christoph, Larcher, Gerhard
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1608.06847

Abstract

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

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