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

TL;DR

This paper investigates the existence of bounded and strongly non-bounded remainder sets in sequences of the form $(\{a_n\alpha\})$, introducing the concept of S-NBRS and analyzing their presence in polynomial and exponential sequences.

Contribution

It introduces the concept of strongly non-bounded remainder sets (S-NBRS) and characterizes their existence for various classes of sequences, including polynomial and exponential types.

Findings

Polynomial-type sequences lack S-NBRS.

Sequences like $(\{2^n\alpha\})$ have every interval as an S-NBRS.

Results extend understanding of distribution properties of specific sequences.

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.