Inhomogeneous Kaufman measures and diophantine approximation

Chow, Sam; Zafeiropoulos, Agamemnon; Zorin, Evgeniy

doi:10.2140/apde.2026.19.1547

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Inhomogeneous Kaufman measures and diophantine approximation

Sam Chow, Agamemnon Zafeiropoulos and Evgeniy Zorin

Vol. 19 (2026), No. 8, 1547–1600

DOI: 10.2140/apde.2026.19.1547

Abstract

We introduce an inhomogeneous variant of Kaufman’s measure for badly approximable numbers. Our measure is supported on the set of pairs $(α, γ)$ such that $α$ is $γ$ -inhomogeneously badly approximable. This allows us to make progress related to an inhomogeneous version of Littlewood’s conjecture.

Keywords

diophantine approximation, symbolic dynamics, Fourier decay of measures

Mathematical Subject Classification

Primary: 11J04, 11J70, 11J71, 11J83, 11J99

Milestones

Received: 18 March 2024

Revised: 3 April 2025

Accepted: 2 November 2025

Published: 8 October 2026

Authors

Sam Chow
Mathematics Institute University of Warwick Coventry United Kingdom

Agamemnon Zafeiropoulos
Department of Mathematics Technion Haifa Israel	TU Graz Graz Austria

Evgeniy Zorin
Department of Mathematics University of York York United Kingdom

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Vol. 19, No. 8, 2026