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Abstract

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.

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N.A.-Research supported in part by a U.S.A.-Israel BSF grant.

Y.P.-Partially supported by a Weizmann Postdoctoral Fellowship.

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Alon, N., Peres, Y. Uniform dilations. Geometric and Functional Analysis 2, 1–28 (1992). https://doi.org/10.1007/BF01895704

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  • DOI: https://doi.org/10.1007/BF01895704

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