Point-Line Incidence Estimates in $(\mathbb{Z}/p^k\mathbb{Z})^2$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909864825978880 |
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| author | Chu, Yuhan |
| author_facet | Chu, Yuhan |
| contents | The point-line incidence problem has been widely studied in Euclidean spaces and vector spaces over finite fields, whereas the analogous problem has rarely been considered over finite $p$-adic rings. In this paper, we investigate incidences in the $p$-adic setting and prove new incidence bounds for points and lines in $(\mathbb{Z}/p^k\mathbb{Z})^2$. Our first two results extend previously known incidence bounds over finite fields, assuming lines are well-separated. For non-separated lines, we establish a general incidence result for weighted points and lines under certain dimensional spacing conditions using the Fourier analytic method and the induction-on-scales argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20104 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Point-Line Incidence Estimates in $(\mathbb{Z}/p^k\mathbb{Z})^2$ Chu, Yuhan Combinatorics The point-line incidence problem has been widely studied in Euclidean spaces and vector spaces over finite fields, whereas the analogous problem has rarely been considered over finite $p$-adic rings. In this paper, we investigate incidences in the $p$-adic setting and prove new incidence bounds for points and lines in $(\mathbb{Z}/p^k\mathbb{Z})^2$. Our first two results extend previously known incidence bounds over finite fields, assuming lines are well-separated. For non-separated lines, we establish a general incidence result for weighted points and lines under certain dimensional spacing conditions using the Fourier analytic method and the induction-on-scales argument. |
| title | Point-Line Incidence Estimates in $(\mathbb{Z}/p^k\mathbb{Z})^2$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.20104 |