Bohr sets and multiplicative diophantine approximation

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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.
Original languageEnglish
Pages (from-to)1623-1642
Number of pages20
JournalDuke Mathematical Journal
Issue number9
Early online date23 Mar 2018
Publication statusPublished - 15 Jun 2018

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  • metric diophantine approximation
  • additive combinatorics
  • geometry of numbers

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