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

Prasuna Bandi; Reynold Fregoli; Dmitry Kleinbock

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arXiv · 2504.02258

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

Prasuna Bandi·Reynold Fregoli·Dmitry Kleinbock

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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.

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BibTeX RIS

Prasuna Bandi, Reynold Fregoli, Dmitry Kleinbock. 2025-04-03. Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation. https://arxiv.org/abs/2504.02258

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