### Abstract

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 language | English (US) |
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Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Geometric and Functional Analysis |

Volume | 2 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 1992 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

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## Cite this

Alon, N. M., & Peres, Y. (1992). Uniform dilations.

*Geometric and Functional Analysis*,*2*(1), 1-28. https://doi.org/10.1007/BF01895704