Polynomial Szemerédi for sets with large Hausdorff dimension on the Torus
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916851554975744 |
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| author | Hong, Guo-Dong |
| author_facet | Hong, Guo-Dong |
| contents | Let $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{R}[y]\}$ be a collection of polynomials with distinct degrees and zero constant terms. We proved that there exists $ε=ε(\mathbb{P})>0$ such that, for any compact set $E \subset \mathbb{T}$ with dim(E)$>1-ε$, we can find $y\neq 0$ so that $\{x,x+P_1(y), \cdots,x+P_k(y)\} \subset E$. The proof relies on a suitable version of the Sobolev smoothing inequality with ideas adapted from Peluse \cite{P19}, Durcik and Roos \cite{DR24}, and Krause, Mirek, Peluse, and Wright \cite{KMPW24}. As a byproduct of our Sobolev smoothing inequality, we demonstrated that the divergence set of the pointwise convergence problem for certain polynomial multiple ergodic averages has Hausdorff dimension strictly less than one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial Szemerédi for sets with large Hausdorff dimension on the Torus Hong, Guo-Dong Classical Analysis and ODEs Let $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{R}[y]\}$ be a collection of polynomials with distinct degrees and zero constant terms. We proved that there exists $ε=ε(\mathbb{P})>0$ such that, for any compact set $E \subset \mathbb{T}$ with dim(E)$>1-ε$, we can find $y\neq 0$ so that $\{x,x+P_1(y), \cdots,x+P_k(y)\} \subset E$. The proof relies on a suitable version of the Sobolev smoothing inequality with ideas adapted from Peluse \cite{P19}, Durcik and Roos \cite{DR24}, and Krause, Mirek, Peluse, and Wright \cite{KMPW24}. As a byproduct of our Sobolev smoothing inequality, we demonstrated that the divergence set of the pointwise convergence problem for certain polynomial multiple ergodic averages has Hausdorff dimension strictly less than one. |
| title | Polynomial Szemerédi for sets with large Hausdorff dimension on the Torus |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2507.14407 |