Decorrelation estimates for translated measures under diagonal flows

Michael Björklund; Reynold Fregoli; Alexander Gorodnik

doi:10.3934/jmd.2025007

Article Contents

2025, Volume 21: 401-441. Doi: 10.3934/jmd.2025007

This volume Previous Article Anosov flows with the same periodic orbits Next Article A global shadow lemma and logarithm law for geometrically finite Hilbert geometries

Decorrelation estimates for translated measures under diagonal flows

1.
Chalmers University, Mathematical Sciences, Chalmers Tvärgata 3, SE-412 96 Gothenburg, Sweden
2.
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA
3.
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland

Received: January 17, 2024

Revised: December 5, 2024

Published online: June 27, 2025

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Abstract

A profound link between homogeneous dynamics and Diophantine approximation is based on an observation that Diophantine properties of a real matrix $ B $ are encoded by the corresponding lattice $ \Lambda_B $ translated by a multi-parameter semigroup $ a(t) $. We establish quantitative decorrelation estimates for measures supported on leaves $ a(t)\Lambda_B $ with the error terms depending only on the minimum of the pairwise distances between the parameters. The proof involves a careful analysis of the translated measures in the products of the spaces of unimodular lattices and establishes quantitative equidistributions to measures supported on various intermediate homogeneous subspaces.

Keywords:

Decorrelation of unipotent orbits,
higher order equidistribution and mixing.

Mathematics Subject Classification: 37A17, 37A25; Secondary: 37A44.

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