Uniform dilations

Every sufficiently large finite set X in [0,1) has a dilation nX mod 1 with small maximal gap and even small discrepancy. We establish a sharp quantitative version of this principle, which puts into a broader perspective some classical results on the distribution of power residues. The proof is based on a second-moment argument which reduces the problem to an estimate on the number of edges in a certain graph. Cycles in this graph correspond to solutions of a simple Diophantine equation: The growth asymptotics of these solutions, which can be determined from properties of lattices in Euclidean space, yield the required estimate.