Non-Salem sets in multiplicative Diophantine approximation
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913507962781696 |
|---|---|
| author | Tan, Bo Zhou, Qing-Long |
| author_facet | Tan, Bo Zhou, Qing-Long |
| contents | In this paper, we answer a question of Cai-Hambrook in (arXiv$\colon$ 2403.19410). Furthermore, we compute the Fourier dimension of the multiplicative $ψ$-well approximable set $$M_2^{\times}(ψ)=\left\{(x_1,x_2)\in [0,1]^{2}\colon \|qx_1\|\|qx_2\|<ψ(q) \text{ for infinitely many } q\in \N\right\},$$ where $ψ\colon\N\to [0,\frac{1}{4})$ is a positive function satisfying $\sum_qψ(q)\log\frac{1}{ψ(q)}<\infty.$ As a corollary, we show that the set $M_2^{\times}(q\mapsto q^{-τ})$ is non-Salem for $τ>1.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12557 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Salem sets in multiplicative Diophantine approximation Tan, Bo Zhou, Qing-Long Number Theory 28A80, 11J83, 11K55 In this paper, we answer a question of Cai-Hambrook in (arXiv$\colon$ 2403.19410). Furthermore, we compute the Fourier dimension of the multiplicative $ψ$-well approximable set $$M_2^{\times}(ψ)=\left\{(x_1,x_2)\in [0,1]^{2}\colon \|qx_1\|\|qx_2\|<ψ(q) \text{ for infinitely many } q\in \N\right\},$$ where $ψ\colon\N\to [0,\frac{1}{4})$ is a positive function satisfying $\sum_qψ(q)\log\frac{1}{ψ(q)}<\infty.$ As a corollary, we show that the set $M_2^{\times}(q\mapsto q^{-τ})$ is non-Salem for $τ>1.$ |
| title | Non-Salem sets in multiplicative Diophantine approximation |
| topic | Number Theory 28A80, 11J83, 11K55 |
| url | https://arxiv.org/abs/2409.12557 |