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A METRIC THEORY OF MINIMAL GAPS

Published online by Cambridge University Press:  21 May 2018

Zeév Rudnick*
Affiliation:
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel email [email protected]
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Abstract

We study the minimal gap statistic for fractional parts of sequences of the form ${\mathcal{A}}^{\unicode[STIX]{x1D6FC}}=\{\unicode[STIX]{x1D6FC}a(n)\}$, where ${\mathcal{A}}=\{a(n)\}$ is a sequence of distinct integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\unicode[STIX]{x1D6FC}$, the minimal gap $\unicode[STIX]{x1D6FF}_{\min }^{\unicode[STIX]{x1D6FC}}(N)=\min \{\unicode[STIX]{x1D6FC}a(m)-\unicode[STIX]{x1D6FC}a(n)\hspace{0.2em}{\rm mod}\hspace{0.2em}1:1\leqslant m\neq n\leqslant N\}$ is close to that of a random sequence.

Type
Research Article
Copyright
Copyright © University College London 2018 

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