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THERE IS NO KHINTCHINE THRESHOLD FOR METRIC PAIR CORRELATIONS

Part of: Combinatorics Sequences and sets Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  23 July 2019

Christoph Aistleitner ,
Thomas Lachmann  and
Niclas Technau
Christoph Aistleitner
Affiliation:
Institute for Analysis and Number Theory, University of Technology Graz, Steyrergasse 30, 8010 Graz, Austria email aistleitner@math.tugraz.at
Thomas Lachmann
Affiliation:
Institute for Analysis and Number Theory, University of Technology Graz, Steyrergasse 30, 8010 Graz, Austria email lachmann@math.tugraz.at
Niclas Technau
Affiliation:
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel email technau@math.tugraz.at
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Abstract

We consider sequences of the form $(a_{n}\unicode[STIX]{x1D6FC})_{n}$ mod 1, where $\unicode[STIX]{x1D6FC}\in [0,1]$ and where $(a_{n})_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all $\unicode[STIX]{x1D6FC}$ in the sense of Lebesgue measure, we say that $(a_{n})_{n}$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_{n})_{n}$. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterizes the metric pair correlation property in terms of the additive energy, similar to Khintchine’s criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_{n})_{n}$ having large additive energy which, however, maintains the metric pair correlation property.

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity 11K60: Diophantine approximation
Secondary: 05A18: Partitions of sets 11B25: Arithmetic progressions 11K06: General theory of distribution modulo $1$
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 929 - 949
Copyright
Copyright © University College London 2019 

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