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Higher-rank Bohr sets and multiplicative diophantine approximation

Published online by Cambridge University Press:  24 September 2019

Sam Chow
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK email sam.chow@maths.ox.ac.uk
Niclas Technau
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK email niclas.technau@york.ac.uk

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.

Type
Research Article
Copyright
© The Authors 2019 

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