On the number of lattice points in thin sectors
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911264714784768 |
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| author | Waxman, Ezra Yesha, Nadav |
| author_facet | Waxman, Ezra Yesha, Nadav |
| contents | On the circle of radius $R$ centred at the origin, consider a ``thin'' sector about the fixed line $y = αx$ with edges given by the lines $y = (α\pm ε) x$, where $ε= ε_R \rightarrow 0$ as $ R \to \infty $. We establish an asymptotic count for $S_α(ε,R)$, the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of $ε$ and on the rationality/irrationality type of $α$. In particular, we demonstrate that if $α$ is Diophantine, then $S_α(ε,R)$ is asymptotic to the area of the sector, so long as $εR^{t} \rightarrow \infty$ for some $ t<2 $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_02060 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the number of lattice points in thin sectors Waxman, Ezra Yesha, Nadav Number Theory 11P21, 11H06 On the circle of radius $R$ centred at the origin, consider a ``thin'' sector about the fixed line $y = αx$ with edges given by the lines $y = (α\pm ε) x$, where $ε= ε_R \rightarrow 0$ as $ R \to \infty $. We establish an asymptotic count for $S_α(ε,R)$, the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of $ε$ and on the rationality/irrationality type of $α$. In particular, we demonstrate that if $α$ is Diophantine, then $S_α(ε,R)$ is asymptotic to the area of the sector, so long as $εR^{t} \rightarrow \infty$ for some $ t<2 $. |
| title | On the number of lattice points in thin sectors |
| topic | Number Theory 11P21, 11H06 |
| url | https://arxiv.org/abs/2305.02060 |