Twisted Diophantine approximation on manifolds
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911429583437824 |
|---|---|
| author | Beresnevich, Victor Simmons, David Velani, Sanju |
| author_facet | Beresnevich, Victor Simmons, David Velani, Sanju |
| contents | In twisted Diophantine approximation, for a fixed $m\times n$ matrix $\boldsymbolα$ one is interested in sets of vectors $\boldsymbolβ\in\mathbb R^m$ such that the system of affine forms $\mathbb R^n \ni \mathbf q \mapsto \boldsymbolα\mathbf q + \boldsymbolβ\in \mathbb R^m$ satisfies some given Diophantine condition. In this paper we introduce the notion of manifolds which are of $\boldsymbolα$-twisted Khintchine type for convergence or divergence. We provide sufficient conditions under which nondegenerate analytic manifolds exhibit this twisted Khintchine-type behaviour. Furthermore, we investigate the intersection properties of the sets of $\boldsymbolα$-twisted badly approximable and well approximable vectors with nondegenerate manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04405 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Twisted Diophantine approximation on manifolds Beresnevich, Victor Simmons, David Velani, Sanju Number Theory Dynamical Systems In twisted Diophantine approximation, for a fixed $m\times n$ matrix $\boldsymbolα$ one is interested in sets of vectors $\boldsymbolβ\in\mathbb R^m$ such that the system of affine forms $\mathbb R^n \ni \mathbf q \mapsto \boldsymbolα\mathbf q + \boldsymbolβ\in \mathbb R^m$ satisfies some given Diophantine condition. In this paper we introduce the notion of manifolds which are of $\boldsymbolα$-twisted Khintchine type for convergence or divergence. We provide sufficient conditions under which nondegenerate analytic manifolds exhibit this twisted Khintchine-type behaviour. Furthermore, we investigate the intersection properties of the sets of $\boldsymbolα$-twisted badly approximable and well approximable vectors with nondegenerate manifolds. |
| title | Twisted Diophantine approximation on manifolds |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2507.04405 |