Uniform dilations |
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Authors: | N. Alon Y. Peres |
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Affiliation: | (1) School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978 Tel Aviv, Israel;(2) Dept. of Mathematics, Stanford University, Stanford, USA;(3) Present address: Mathematics Department, Yale University, 06520 New Haven, Connecticut, USA |
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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.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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Keywords: | Primary: 11K38 Secondary: 11K06 11J13 |
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