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.
|Number of pages||20|
|Journal||Duke Mathematical Journal|
|Early online date||23 Mar 2018|
|Publication status||Published - 15 Jun 2018|
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- metric diophantine approximation
- additive combinatorics
- geometry of numbers