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.
Original language | English |
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Pages (from-to) | 1623-1642 |
Number of pages | 20 |
Journal | Duke Mathematical Journal |
Volume | 167 |
Issue number | 9 |
Early online date | 23 Mar 2018 |
DOIs | |
Publication status | Published - 15 Jun 2018 |
Bibliographical note
This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.Keywords
- metric diophantine approximation
- additive combinatorics
- geometry of numbers