Simultaneous Diophantine approximation to points on the Veronese curve
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912271591014400 |
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| author | Badziahin, Dzmitry |
| author_facet | Badziahin, Dzmitry |
| contents | We compute the Hausdorff dimension of the set of simultaneously $q^{-λ}$-well approximable points on the Veronese curve in $\mathbb{R}^n$ for $λ$ between $\frac{1}{n}$ and $\frac{2}{2n-1}$. For $n=3$, the same result is given for a wider range of $λ$ between $\frac13$ and $\frac12$. We also provide a nontrivial upper bound for this Hausdorff dimension in the case $λ\le \frac{2}{n}$. In the course of the proof we establish that the number of cubic polynomials of height at most $H$ and non-zero discriminant at most $D$ is bounded from above by $c(ε) H^{2/3 + ε} D^{5/6}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17685 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simultaneous Diophantine approximation to points on the Veronese curve Badziahin, Dzmitry Number Theory 11J13, 11J54, 11J82, 11K55 We compute the Hausdorff dimension of the set of simultaneously $q^{-λ}$-well approximable points on the Veronese curve in $\mathbb{R}^n$ for $λ$ between $\frac{1}{n}$ and $\frac{2}{2n-1}$. For $n=3$, the same result is given for a wider range of $λ$ between $\frac13$ and $\frac12$. We also provide a nontrivial upper bound for this Hausdorff dimension in the case $λ\le \frac{2}{n}$. In the course of the proof we establish that the number of cubic polynomials of height at most $H$ and non-zero discriminant at most $D$ is bounded from above by $c(ε) H^{2/3 + ε} D^{5/6}$. |
| title | Simultaneous Diophantine approximation to points on the Veronese curve |
| topic | Number Theory 11J13, 11J54, 11J82, 11K55 |
| url | https://arxiv.org/abs/2403.17685 |