Minimal gap for higher dimensional sequences

Abstract

In this note, we extend the notion of minimal gaps to the higher dimensional sequences. We bound the minimal gap for $(\{{\mathbf{a}}_n\mathbf{\alpha }\}),$ $(\{a_n\mathbf{\alpha }\})$ and $(\{{\mathbf{a}}_n\cdot \mathbf{\alpha }\})$ in terms of cardinality of the difference set of $a_n$ and ${\mathbf{a}}_n,$ where $a_n$ and $a_n^{(1)},\dots ,a_n^{(d)}$ are sequences of distinct integers.

arXiv comment13 pages. Comments are welcome