Bohr sets and multiplicative diophantine approximation

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JournalDuke Mathematical Journal
DateAccepted/In press - 11 Jan 2018
DateE-pub ahead of print - 23 Mar 2018
DatePublished (current) - 15 Jun 2018
Issue number9
Volume167
Pages (from-to)1623-1642
Early online date23/03/18
Original languageEnglish

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 fibre 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 generalised 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.

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    Research areas

  • metric diophantine approximation, additive combinatorics, geometry of numbers

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