Submanifold-genericity of ℝ^d-actions and uniform multiplicative Diophantine approximation

2025/04/03 by Prasuna Bandi, Reynold Fregoli, Bandi, Prasuna +3 · 2 citations
Computer Science · Mathematics · #11J13 #11K60 #37A25 #37A44 #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2504.02258

openalex publication_date 2025/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove a new ergodic theorem for ℝ^d-actions involving averages over dilated submanifolds, thereby generalizing the theory of spherical averages. Our main result is a quantitative estimate for the error term of such averages valid for smooth functions under some effective mixing assumptions on the action. With the aid of this theorem, we investigate multiplicative-type Dirichlet-improvability for (m× n)-matrices with real coefficients. In particular, we establish that almost all matrices are uniformly approximable by the function x↦ x^-1(log x)^-1+ε for any ε>0. Results of this type motivate a question which can be thought as a strengthening of Littlewood's conjecture in multiplicative Diophantine approximation.

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• Simultaneously bounded and dense orbits for commuting Cartan actions
• Higher-Dimensional Moving Averages and Submanifold Genericity

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