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A METRIC THEORY OF MINIMAL GAPS

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  21 May 2018

Zeév Rudnick
Zeév Rudnick*
Affiliation:
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel email rudnick@post.tau.ac.il
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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.

MSC classification

Primary: 11K06: General theory of distribution modulo $1$
Type
Research Article
Information
Mathematika , Volume 64 , Issue 3 , 2018 , pp. 628 - 636
Copyright
Copyright © University College London 2018 

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