Additive Energy and Irregularities of Distribution

Abstract: We consider strictly increasing sequences

l e f t (a_{n} i g h t)_{n g e q 1}

of integers and sequences of fractional parts

l e f t (l e f t a_{n} a l p h a i g h t i g h t)_{n g e q 1}

where

a l p h a i n m a t h b b R

. We show that a small additive energy of

l e f t (a_{n} i g h t)_{n g e q 1}

implies that for almost all

a l p h a

the sequence

l e f t (l e f t a_{n} a l p h a i g h t i g h t)_{n g e q 1}

has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.