Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915225141248000 |
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| author | Bandi, Prasuna Fregoli, Reynold Kleinbock, Dmitry |
| author_facet | Bandi, Prasuna Fregoli, Reynold Kleinbock, Dmitry |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02258 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation Bandi, Prasuna Fregoli, Reynold Kleinbock, Dmitry Number Theory Dynamical Systems 37A25, 37A44, 11J13, 11K60 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. |
| title | Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation |
| topic | Number Theory Dynamical Systems 37A25, 37A44, 11J13, 11K60 |
| url | https://arxiv.org/abs/2504.02258 |