Density of Rational Points Near Flat/Rough Hypersurfaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914887503970304 |
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| author | Srivastava, Rajula Technau, Niclas |
| author_facet | Srivastava, Rajula Technau, Niclas |
| contents | For $n\geq 3$, let $\mathscr{M} \subseteq\mathbb{R}^{n}$ be a compact hypersurface, parametrized by a homogeneous function of degree $d\in \mathbb{R}_{>1}$, with non-vanishing curvature away from the origin. Consider the number $\mathrm{N}_{\mathscr{M}}(δ,Q)$ of rationals $\mathbf{a}/q$, with denominator $q\in [Q,2Q)$ and $\mathbf{a} \in \mathbb{Z}^{n-1}$, lying at a distance at most $δ/q$ from $\mathscr{M}$. This manuscript provides essentially sharp estimates for $\mathrm{N}_{\mathscr{M}}(δ,Q)$ throughout the range $δ\in (Q^{\varepsilon-1},1/2)$ for $d>1+\tfrac{1}{2n-3}$. Our result is a first of its kind for hypersurfaces with vanishing Gaussian curvature ($d>2$) and those which are rough (meaning not even $C^2$ at the origin which happens when $d<2$). An interesting outcome of our investigation is the understanding of a `geometric' term $(δ/Q)^{(n-1)/d}Q^n$ (stemming from a so-called Knapp cap), arising in addition to the usual probabilistic term $δQ^n$; the sum of these terms determines the size of $\mathrm{N}_{\mathscr{M}}(δ,Q)$ for $δ\in(Q^{\varepsilon-1},1/2)$. Consequences of our result concern the metric theory of Diophantine approximation on `rough' hypersurfaces -- going beyond the recent break-through of Beresnevich and L. Yang. Further, we establish smooth extensions of Serre's dimension growth conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_01047 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Density of Rational Points Near Flat/Rough Hypersurfaces Srivastava, Rajula Technau, Niclas Number Theory Classical Analysis and ODEs 11J83, 11D75, 42B99 For $n\geq 3$, let $\mathscr{M} \subseteq\mathbb{R}^{n}$ be a compact hypersurface, parametrized by a homogeneous function of degree $d\in \mathbb{R}_{>1}$, with non-vanishing curvature away from the origin. Consider the number $\mathrm{N}_{\mathscr{M}}(δ,Q)$ of rationals $\mathbf{a}/q$, with denominator $q\in [Q,2Q)$ and $\mathbf{a} \in \mathbb{Z}^{n-1}$, lying at a distance at most $δ/q$ from $\mathscr{M}$. This manuscript provides essentially sharp estimates for $\mathrm{N}_{\mathscr{M}}(δ,Q)$ throughout the range $δ\in (Q^{\varepsilon-1},1/2)$ for $d>1+\tfrac{1}{2n-3}$. Our result is a first of its kind for hypersurfaces with vanishing Gaussian curvature ($d>2$) and those which are rough (meaning not even $C^2$ at the origin which happens when $d<2$). An interesting outcome of our investigation is the understanding of a `geometric' term $(δ/Q)^{(n-1)/d}Q^n$ (stemming from a so-called Knapp cap), arising in addition to the usual probabilistic term $δQ^n$; the sum of these terms determines the size of $\mathrm{N}_{\mathscr{M}}(δ,Q)$ for $δ\in(Q^{\varepsilon-1},1/2)$. Consequences of our result concern the metric theory of Diophantine approximation on `rough' hypersurfaces -- going beyond the recent break-through of Beresnevich and L. Yang. Further, we establish smooth extensions of Serre's dimension growth conjecture. |
| title | Density of Rational Points Near Flat/Rough Hypersurfaces |
| topic | Number Theory Classical Analysis and ODEs 11J83, 11D75, 42B99 |
| url | https://arxiv.org/abs/2305.01047 |