Uniform dilations

N. Alon*, Y. Peres

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalGeometric and Functional Analysis
Issue number1
StatePublished - Mar 1992


  • AMS Classification: Primary: 11K38, Secondary: 11K06, 11J13
  • Discrepancy
  • density mod 1
  • dilation
  • second moment method


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