Higher-rank Bohr sets and multiplicative diophantine approximation

Sam Chow, Niclas Technau

Research output: Contribution to journalArticlepeer-review


Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.

Original languageEnglish
Pages (from-to)2214-2233
Number of pages20
JournalCompositio Mathematica
Issue number11
StatePublished - 1 Nov 2019
Externally publishedYes


  • Additive combinatorics
  • Geometry of numbers
  • Metric diophantine approximation


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