Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910923036295168 |
|---|---|
| author | Tan, Bo Zhou, Qing-Long |
| author_facet | Tan, Bo Zhou, Qing-Long |
| contents | Let $E\subset [0,1)^{d}$ be a set supporting a probability measure $μ$ with Fourier decay $|\widehatμ({\bf{t}})|\ll (\log |{\bf{t}}|)^{-s}$ for some constant $s>d+1.$ Consider a sequence of expanding integral matrices $\mathcal{A}=(A_n)_{n\in\N}$ such that the minimal singular values of $A_{n+1}A_{n}^{-1}$ are uniformly bounded below by $K>1$. We prove a quantitative Schmidt-type counting theorem under the following constraints: (1) the points of interest are restricted to $E$; (2) the denominators of the ``shifted'' rational approximations are drawn exclusively from $\mathcal{A}$. Our result extends the work of Pollington, Velani, Zafeiropoulos, and Zorin (2022) to the matrix setting, advancing the study of Diophantine approximation on fractals. Moreover, it strengthens the equidistribution property of the sequence $(A_n{\bf x})_{n\in\N}$ for $μ$-almost every ${\bf x}\in E.$ Applications include the normality of vectors and shrinking target problems on fractal sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21555 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets Tan, Bo Zhou, Qing-Long Number Theory 28A80, 11J83 Let $E\subset [0,1)^{d}$ be a set supporting a probability measure $μ$ with Fourier decay $|\widehatμ({\bf{t}})|\ll (\log |{\bf{t}}|)^{-s}$ for some constant $s>d+1.$ Consider a sequence of expanding integral matrices $\mathcal{A}=(A_n)_{n\in\N}$ such that the minimal singular values of $A_{n+1}A_{n}^{-1}$ are uniformly bounded below by $K>1$. We prove a quantitative Schmidt-type counting theorem under the following constraints: (1) the points of interest are restricted to $E$; (2) the denominators of the ``shifted'' rational approximations are drawn exclusively from $\mathcal{A}$. Our result extends the work of Pollington, Velani, Zafeiropoulos, and Zorin (2022) to the matrix setting, advancing the study of Diophantine approximation on fractals. Moreover, it strengthens the equidistribution property of the sequence $(A_n{\bf x})_{n\in\N}$ for $μ$-almost every ${\bf x}\in E.$ Applications include the normality of vectors and shrinking target problems on fractal sets. |
| title | Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets |
| topic | Number Theory 28A80, 11J83 |
| url | https://arxiv.org/abs/2504.21555 |