Quantitative bounds in a popular polynomial Szemerédi theorem
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908642544975872 |
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| author | Shao, Xuancheng Wang, Mengdi |
| author_facet | Shao, Xuancheng Wang, Mengdi |
| contents | We obtain polylogarithmic bounds in the polynomial Szemerédi theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each having zero constant term. Then there exists a constant $c = c(P_1,\dots,P_m) > 0$ such that any subset $A \subset \{1,2,\dots,N\}$ of density at least $(\log N)^{-c}$ contains a nontrivial polynomial progression of the form $x, x+P_1(y), \dots, x+P_m(y)$. In addition, we prove an effective ``popular'' version, showing that every dense subset $A$ has some non-zero $y$ such that the number of polynomial progressions in $A$ with this difference $y$ is asymptotically at least as large as in a random set of the same density as $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16822 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative bounds in a popular polynomial Szemerédi theorem Shao, Xuancheng Wang, Mengdi Number Theory Combinatorics We obtain polylogarithmic bounds in the polynomial Szemerédi theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each having zero constant term. Then there exists a constant $c = c(P_1,\dots,P_m) > 0$ such that any subset $A \subset \{1,2,\dots,N\}$ of density at least $(\log N)^{-c}$ contains a nontrivial polynomial progression of the form $x, x+P_1(y), \dots, x+P_m(y)$. In addition, we prove an effective ``popular'' version, showing that every dense subset $A$ has some non-zero $y$ such that the number of polynomial progressions in $A$ with this difference $y$ is asymptotically at least as large as in a random set of the same density as $A$. |
| title | Quantitative bounds in a popular polynomial Szemerédi theorem |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2505.16822 |