Effective density of values of indefinite ternary inhomogeneous quadratic forms
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909280601374720 |
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| author | Kelmer, Dubi |
| author_facet | Kelmer, Dubi |
| contents | Given an inhomogeneous quadratic form $Q_ξ(v)=Q(v+ξ)$ with $Q$ an indefinite $\mathbb{Q}$-isotropic rational ternary form and $ξ\in \mathbb{R}^3$ irrational, we prove an effective lower bound for the number of integer vectors $v\in \mathbb{Z}^n$ with $\|v\| \leq T$ such that $|Q_ξ(v)-t|<δ$ that is valid for any $t\in \mathbb{R}$ and all $δ\geq T^{-ν}$, with $ν>0$ depending explicitly on the Diophantine properties of $ξ$. In particular, for $ξ$ with algebraic entries we can take any $ν<\frac{1}{8}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Effective density of values of indefinite ternary inhomogeneous quadratic forms Kelmer, Dubi Number Theory Given an inhomogeneous quadratic form $Q_ξ(v)=Q(v+ξ)$ with $Q$ an indefinite $\mathbb{Q}$-isotropic rational ternary form and $ξ\in \mathbb{R}^3$ irrational, we prove an effective lower bound for the number of integer vectors $v\in \mathbb{Z}^n$ with $\|v\| \leq T$ such that $|Q_ξ(v)-t|<δ$ that is valid for any $t\in \mathbb{R}$ and all $δ\geq T^{-ν}$, with $ν>0$ depending explicitly on the Diophantine properties of $ξ$. In particular, for $ξ$ with algebraic entries we can take any $ν<\frac{1}{8}$. |
| title | Effective density of values of indefinite ternary inhomogeneous quadratic forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.03153 |