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Geometric and Functional Analysis Vol. 2, No. 1 (1992)
 

Summary: Geometric and Functional Analysis
Vol. 2, No. 1 (1992)
1016-443X/92/01001-2851.50+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 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.
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 c-dense 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

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics