Good functions, measures, and the Kleinbock-Tomanov conjecture
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910903704748032 |
|---|---|
| author | Beresnevich, Victor Datta, Shreyasi Ghosh, Anish |
| author_facet | Beresnevich, Victor Datta, Shreyasi Ghosh, Anish |
| contents | In this paper we prove a conjecture of Kleinbock and Tomanov \cite[Conjecture~FP]{KT} on Diophantine properties of a large class of fractal measures on $\mathbb{Q}_p^n$. More generally, we establish the $p$-adic analogues of the influential results of Kleinbock, Lindenstrauss, and Weiss \cite{KLW} on Diophantine properties of friendly measures. We further prove the $p$-adic analogue of one of the main results in \cite{Kleinbock-exponent} due to Kleinbock concerning Diophantine inheritance of affine subspaces, which answers a question of Kleinbock. One of the key ingredients in the proofs of \cite{KLW} is a result on $(C, α)$-good functions whose proof crucially uses the mean value theorem. Our main technical innovation is an alternative approach to establishing that certain functions are $(C, α)$-good in the $p$-adic setting. We believe this result will be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_10456 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Good functions, measures, and the Kleinbock-Tomanov conjecture Beresnevich, Victor Datta, Shreyasi Ghosh, Anish Number Theory Dynamical Systems 11J61, 37A17 In this paper we prove a conjecture of Kleinbock and Tomanov \cite[Conjecture~FP]{KT} on Diophantine properties of a large class of fractal measures on $\mathbb{Q}_p^n$. More generally, we establish the $p$-adic analogues of the influential results of Kleinbock, Lindenstrauss, and Weiss \cite{KLW} on Diophantine properties of friendly measures. We further prove the $p$-adic analogue of one of the main results in \cite{Kleinbock-exponent} due to Kleinbock concerning Diophantine inheritance of affine subspaces, which answers a question of Kleinbock. One of the key ingredients in the proofs of \cite{KLW} is a result on $(C, α)$-good functions whose proof crucially uses the mean value theorem. Our main technical innovation is an alternative approach to establishing that certain functions are $(C, α)$-good in the $p$-adic setting. We believe this result will be of independent interest. |
| title | Good functions, measures, and the Kleinbock-Tomanov conjecture |
| topic | Number Theory Dynamical Systems 11J61, 37A17 |
| url | https://arxiv.org/abs/2209.10456 |