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 (ₙα\), (ₙα\) and (ₙ·α\) in terms of cardinality of the difference set of aₙ and aₙ, where aₙ and aₙ⁽¹⁾,,aₙ⁽ᵈ⁾ are sequences of distinct integers.