Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914938154385408 |
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| author | Tan, Bo Zhou, Qing-Long |
| author_facet | Tan, Bo Zhou, Qing-Long |
| contents | Let $E\subset [0,1]$ be a set that supports a probability measure $μ$ with the property that $|\widehatμ(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence. We present a quantitative inhomogeneous Khintchine-type theorem in which the points of interest are restricted to $E$ and the denominators of the shifted fractions are restricted to $\mathcal{A}.$ Our result improves and extends a previous result in this direction obtained by Pollington-Velani-Zafeiropoulos-Zorin (2022). We also show that the Dirichlet non-improvable set VS well-approximable set is of positive Fourier dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03331 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers Tan, Bo Zhou, Qing-Long Number Theory 28A80, 11K55, 11J83 Let $E\subset [0,1]$ be a set that supports a probability measure $μ$ with the property that $|\widehatμ(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence. We present a quantitative inhomogeneous Khintchine-type theorem in which the points of interest are restricted to $E$ and the denominators of the shifted fractions are restricted to $\mathcal{A}.$ Our result improves and extends a previous result in this direction obtained by Pollington-Velani-Zafeiropoulos-Zorin (2022). We also show that the Dirichlet non-improvable set VS well-approximable set is of positive Fourier dimension. |
| title | Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers |
| topic | Number Theory 28A80, 11K55, 11J83 |
| url | https://arxiv.org/abs/2409.03331 |