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 $(\{\boldsymbol{a}_n\boldsymbolα\}),$ $(\{a_n\boldsymbolα\})$ and $(\{\boldsymbol{a}_n\cdot\boldsymbolα\})$ in terms of cardinality of the difference set of $a_n$ and $\boldsymbol{a}_n,$ where $a_n$ and $a_n^{(1)},\dots,a_n^{(d)}$ are sequences of distinct integers.