Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.22512 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915817952641024 |
|---|---|
| author | Tapia, Lucas |
| author_facet | Tapia, Lucas |
| contents | We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers $\{a_n\}_{n\in\mathbb{N}},\, \{b_n\}_{n\in\mathbb{N}},\, \{c_n\}_{n\in\mathbb{N}},\, \{d_n\}_{n\in\mathbb{N}},$ we determine convergence conditions under which the set of $x\in [0,1]$ which satisfy $\left\lVert a_n x +c_n\right\rVert \cdot \left\lVert b_n x + d_n \right\rVert < ψ(n) $ for infinitely many $n\in\mathbb{N}$ has zero Hausdorff s-measure. We also obtain an upper bound for the Hausdorff dimension in the inhomogeneous setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22512 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiplicative Diophantine Approximation on Planar Lines with Restricted Denominators Tapia, Lucas Number Theory 11K60 We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers $\{a_n\}_{n\in\mathbb{N}},\, \{b_n\}_{n\in\mathbb{N}},\, \{c_n\}_{n\in\mathbb{N}},\, \{d_n\}_{n\in\mathbb{N}},$ we determine convergence conditions under which the set of $x\in [0,1]$ which satisfy $\left\lVert a_n x +c_n\right\rVert \cdot \left\lVert b_n x + d_n \right\rVert < ψ(n) $ for infinitely many $n\in\mathbb{N}$ has zero Hausdorff s-measure. We also obtain an upper bound for the Hausdorff dimension in the inhomogeneous setting. |
| title | Multiplicative Diophantine Approximation on Planar Lines with Restricted Denominators |
| topic | Number Theory 11K60 |
| url | https://arxiv.org/abs/2602.22512 |