Geometric & Functional Analysis GAFA

, Volume 2, Issue 1, pp 1–28 | Cite as

Uniform dilations

  • N. Alon
  • Y. Peres


Every sufficiently large finite setX in [0,1) has a dilationnX 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.

AMS Classification

Primary: 11K38 Secondary: 11K06 11J13 


Discrepancy density mod 1 dilation second moment method 


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Copyright information

© Birkhäuser Verlag 1992

Authors and Affiliations

  • N. Alon
    • 1
  • Y. Peres
    • 2
  1. 1.School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Dept. of MathematicsStanford UniversityStanfordUSA

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