  We provide bounds on the sizes of the gaps -- defined broadly -- in the set
$\{k_1\vbeta_1 + \ldots + k_n\vbeta_n \mbox{ (mod 1)} : k_i \in \Z \cap
(0,Q^\frac{1}{n}]\}$ for generic $\vbeta_1, \ldots, \vbeta_n \in \R^m$ and all
sufficiently large $Q$. We also introduce a related problem in Diophantine
approximation, which we believe is of independent interest.
