Minimal gap for higher dimensional sequences

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