A multidimensional Szemerédi theorem in integers
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911657653960704 |
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| author | Guo, Jingwei Miao, Changxing Zhan, Guoqing |
| author_facet | Guo, Jingwei Miao, Changxing Zhan, Guoqing |
| contents | For any integer $n \geq 2$, let $(m_{1},\ldots,m_{n})$ be a strictly increasing $n$-tuple of positive integers. We show that any subset $A\subset [N]^n$ of density at least $(\log N)^{-c}$ contains a nontrivial configuration of the form \begin{equation*} \boldsymbol{x},\boldsymbol{x}+r^{m_{1}}\boldsymbol{e_{1}},\ldots,\boldsymbol{x}+r^{m_{n}}\boldsymbol{e_{n}}, \end{equation*} where $c=c(n,m_{1},\ldots,m_{n} )$ is a positive constant. This quantitative multidimensional Szemerédi theorem extends a recent two-dimensional result of Peluse, Prendiville, and Shao concerning the configuration of the form $(x,y),(x+r,y),\left(x,y+r^{2}\right)$. The theorem is obtained as a consequence of an effective ``popular'' version. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06360 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A multidimensional Szemerédi theorem in integers Guo, Jingwei Miao, Changxing Zhan, Guoqing Number Theory Classical Analysis and ODEs Combinatorics For any integer $n \geq 2$, let $(m_{1},\ldots,m_{n})$ be a strictly increasing $n$-tuple of positive integers. We show that any subset $A\subset [N]^n$ of density at least $(\log N)^{-c}$ contains a nontrivial configuration of the form \begin{equation*} \boldsymbol{x},\boldsymbol{x}+r^{m_{1}}\boldsymbol{e_{1}},\ldots,\boldsymbol{x}+r^{m_{n}}\boldsymbol{e_{n}}, \end{equation*} where $c=c(n,m_{1},\ldots,m_{n} )$ is a positive constant. This quantitative multidimensional Szemerédi theorem extends a recent two-dimensional result of Peluse, Prendiville, and Shao concerning the configuration of the form $(x,y),(x+r,y),\left(x,y+r^{2}\right)$. The theorem is obtained as a consequence of an effective ``popular'' version. |
| title | A multidimensional Szemerédi theorem in integers |
| topic | Number Theory Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2605.06360 |