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Higher-rank Bohr sets and multiplicative diophantine approximation

Part of: Geometry of numbers Discrete geometry Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  24 September 2019

Sam Chow  and
Niclas Technau
Sam Chow
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK email [email protected]
Niclas Technau
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK email [email protected]
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Abstract

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.

Keywords

metric diophantine approximationgeometry of numbersadditive combinatorics

MSC classification

Primary: 11J83: Metric theory 11J20: Inhomogeneous linear forms 11H06: Lattices and convex bodies 52C07: Lattices and convex bodies in $n$ dimensions
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 11 , November 2019 , pp. 2214 - 2233
Copyright
© The Authors 2019 

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