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

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.