On Bounded Remainder Sets and Strongly Non-Bounded Remainder Sets for Sequences $(\{a_nα\})_{n\geq 1}$

Abstract: We give some results on the existence of bounded remainder sets (BRS) for sequences of the form $({a_n\alpha}){n\geq 1}$, where $(a_n){n\geq 1}$ - in most cases - is a given sequence of distinct integers. Further we introduce the concept of strongly non-bounded remainder sets (S-NBRS) and we show for a very general class of polynomial-type sequences that these sequences cannot have any S-NBRS, whereas for the sequence $({2^n\alpha})_{n \geq 1}$ every interval is an S-NBRS.