December 2020 The metric theory of the pair correlation function of real-valued lacunary sequences
Zeév Rudnick, Niclas Technau
Illinois J. Math. 64(4): 583-594 (December 2020). DOI: 10.1215/00192082-8720506

Abstract

Let { a ( x ) } x = 1 be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations α a ( x ) is Poissonian for Lebesgue almost every α R . By using harmonic analysis, our result—irrespective of the choice of the real-valued sequence { a ( x ) } x = 1 —can essentially be reduced to showing that the number of solutions to the Diophantine inequality | n 1 ( a ( x 1 ) a ( y 1 ) ) n 2 ( a ( x 2 ) a ( y 2 ) ) | < 1 in integer six-tuples ( n 1 , n 2 , x 1 , x 2 , y 1 , y 2 ) located in the box [ N , N ] 6 with the “excluded diagonals”; that is, x 1 y 1 , x 2 y 2 , ( n 1 , n 2 ) ( 0 , 0 ) , is at most N 4 δ for some fixed δ > 0 , for all sufficiently large N .

Citation

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Zeév Rudnick. Niclas Technau. "The metric theory of the pair correlation function of real-valued lacunary sequences." Illinois J. Math. 64 (4) 583 - 594, December 2020. https://doi.org/10.1215/00192082-8720506

Information

Received: 4 February 2020; Revised: 10 June 2020; Published: December 2020
First available in Project Euclid: 16 September 2020

zbMATH: 07269221
MathSciNet: MR4164447
Digital Object Identifier: 10.1215/00192082-8720506

Subjects:
Primary: 11J54
Secondary: 11J71

Rights: Copyright © 2020 University of Illinois at Urbana-Champaign

Vol.64 • No. 4 • December 2020
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