  In this note, we extend the notion of minimal gaps to the higher dimensional
sequences. We bound the minimal gap for
$(\{\boldsymbol{a}_n\boldsymbol{\alpha}\}),$ $(\{a_n\boldsymbol{\alpha}\})$ and
$(\{\boldsymbol{a}_n\cdot\boldsymbol{\alpha}\})$ 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.
