  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.
