Bounds on the gaps in the fractional parts of a linear form

Abstract

We provide bounds on the sizes of the gaps -- defined broadly -- in the set $\{k_1\beta_1 + \ldots + k_n\beta_n \mbox{ (mod 1)} : k_i \in \mathbb Z \cap (0,Q^\frac{1}{n}]\}$ for generic $\beta_1, \ldots, \beta_n \in \mathbb R^m$ and all sufficiently large $Q$. We also introduce a related problem in Diophantine approximation, which we believe is of independent interest.