Additive Energy and Irregularities of Distribution

We consider strictly increasing sequences (a_n)_n≥1 of integers and sequences of fractional parts ({a_nα})_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_nα})_n≥1 has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.