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15 June 2018 Bohr sets and multiplicative Diophantine approximation
Sam Chow
Duke Math. J. 167(9): 1623-1642 (15 June 2018). DOI: 10.1215/00127094-2018-0001

Abstract

In two dimensions, Gallagher’s theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fiber version of Gallagher’s theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes, and Velani. The idea is to find large generalized arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin–Schaeffer theorem for the problem at hand, via the geometry of numbers.

Citation

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Sam Chow. "Bohr sets and multiplicative Diophantine approximation." Duke Math. J. 167 (9) 1623 - 1642, 15 June 2018. https://doi.org/10.1215/00127094-2018-0001

Information

Received: 20 March 2017; Revised: 3 December 2017; Published: 15 June 2018
First available in Project Euclid: 23 March 2018

zbMATH: 06904636
MathSciNet: MR3813593
Digital Object Identifier: 10.1215/00127094-2018-0001

Subjects:
Primary: 11J83
Secondary: 11H06, 11J20, 52C05

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 9 • 15 June 2018
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