 
Summary: Geometric and Functional Analysis
Vol. 2, No. 1 (1992)
1016443X/92/010012851.50+0.20/0
r 1992 Birkh&user Verlag, Basel
UNIFORM DILATIONS
N. ALON AND Y. PERES
Abstract
Every sufficiently large finite set X in [0,1) has a dilation nX rood1 with
small maximal gap and even small discrepancy. We establish a sharp
quantitative version of this principle, which puts into a broader perspec
tive some classical results on the distribution of power residues. The proof
is based on a secondmoment 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.
1. Introduction
Let T = R/Z denote the one dimensional torus, i.e., the set of real numbers
modulo 1. A subset X of T is called cdense if it intersects every interval of
length c in T. A dilation of X is a subset nX  {nx : x E X} where n is an
