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Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

Prasuna Bandi, Reynold Fregoli, Dmitry Kleinbock

TL;DR

The paper proves a new ergodic theorem for ${\mathbb R}^{d}$-actions with averages over dilated submanifolds, yielding a quantitative error term under effective mixing. It then applies this abstract framework to multiplicative Diophantine approximation for ${m\times n}$ matrices, showing that almost all matrices are uniformly approximable by $x\mapsto x^{-1}(\log x)^{-1+\varepsilon}$ and connecting these results to Littlewood-type questions via dynamics on the space of lattices. The core methodology combines moment estimates and Borel–Cantelli arguments with a dynamical Dani correspondence, translating Diophantine properties into homogeneous-space dynamics and enabling Khintchine-type statements in the uniform setting. These contributions establish a robust link between submanifold-dilations, mixing rates, and uniform multiplicative approximation, with potential implications for conjectures in Diophantine approximation and uniform Dirichlet-type results.

Abstract

In this paper, we prove a new ergodic theorem for $\mathbb{R}^d$-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 $(m\times n)$-matrices with real coefficients. In particular, we establish that almost all matrices are uniformly approximable by the function $x\mapsto x^{-1}(\log x)^{-1+\varepsilon}$ for any $\varepsilon>0$. Results of this type motivate a question which can be thought as a strengthening of Littlewood's conjecture in multiplicative Diophantine approximation.

Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

TL;DR

The paper proves a new ergodic theorem for -actions with averages over dilated submanifolds, yielding a quantitative error term under effective mixing. It then applies this abstract framework to multiplicative Diophantine approximation for matrices, showing that almost all matrices are uniformly approximable by and connecting these results to Littlewood-type questions via dynamics on the space of lattices. The core methodology combines moment estimates and Borel–Cantelli arguments with a dynamical Dani correspondence, translating Diophantine properties into homogeneous-space dynamics and enabling Khintchine-type statements in the uniform setting. These contributions establish a robust link between submanifold-dilations, mixing rates, and uniform multiplicative approximation, with potential implications for conjectures in Diophantine approximation and uniform Dirichlet-type results.

Abstract

In this paper, we prove a new ergodic theorem for -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 -matrices with real coefficients. In particular, we establish that almost all matrices are uniformly approximable by the function for any . Results of this type motivate a question which can be thought as a strengthening of Littlewood's conjecture in multiplicative Diophantine approximation.

Paper Structure

This paper contains 23 sections, 22 theorems, 174 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Averages along dilates of submanifolds
  3. Uniform approximation in the multiplicative set-up
  4. Outline of the paper
  5. An Abstract Theorem and Proofs of Theorems \ref{['thm:gmodgamma']} and \ref{['cor:diophantine']}
  6. Two more results
  7. Proof of Theorem \ref{['thm:gmodgamma']} assuming Theorem \ref{['thm:2']}
  8. Proof of Theorem \ref{['cor:diophantine']} assuming Theorem \ref{['thm:2']}
  9. Proof of Theorems \ref{['thm:genericgeneral']} and \ref{['thm:2']}
  10. A covering lemma
  11. Proof of Lemma \ref{['lem:4mom']}
  12. Completion of the proof of Theorem \ref{['thm:genericgeneral']}
  13. Proof of Theorem \ref{['thm:2']}
  14. Correspondence with Dynamics and Proof of Theorem \ref{['thm:special']}
  15. The multiplicative Dani Correspondence
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $G$ be a semi-simple Lie group of real rank $d$ whose simple factors have rank at least $2$. Let $\Gamma$ be a lattice in $G$ and let $\mu$ denote the unique $G$-invariant Radon probability measure on the space $X:=G/\Gamma$. Fix a Cartan subgroup $A$ of $G$, put $\mathfrak a := \operatorname{Li Finally, let $\mathcal{F}:=\mathscr{C}^{\infty}_c(X)$ denote the space of compactly supported smoot

Figures (1)

  • Figure 1: Sets of interest in Diophantine approximation and their relation.

Theorems & Definitions (36)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • Theorem 2.1
  • Theorem 2.2
  • ...and 26 more