Szemerédi's Theorem Along Cantor Sets of Integers
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917277805314048 |
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| author | Burgin, Alex Fragkos, Anastasios Lacey, Michael T. Mena, Dario Reguera, Maria Carmen |
| author_facet | Burgin, Alex Fragkos, Anastasios Lacey, Michael T. Mena, Dario Reguera, Maria Carmen |
| contents | Let $\mathcal C= \{k_1<k_2 < \cdots\}$ be Cantor set of integers, that is a set of integers with restricted digits modulo a base $b$, and suppose $0$ is one of the restricted digits. We show that $$ \liminf_N \Expectation_{n\in [N]} m(A\cap T^{-k_n} A \cap \cdots
\cap T^{-\ell k_n} A )>0. $$ This is an extension of the IP Ergodic Theorem of Furstenberg and Katznelson, and a partial extension of recent work of Kra and Shalom. In particular, this implies that for any subset of integers $A$ of positive upper Banach density, there is a set $B$ of integers $n$ of positive lower Banach density such that $A$ contains an $\ell+1$ term progression, with step size $k_n$, where $n\in B$. This is a complement to recent results of Kra and Shalom, for IP Sets of integers, and Burgin, concerning Sarkozy's Theorem for Primes with restricted digits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15299 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Szemerédi's Theorem Along Cantor Sets of Integers Burgin, Alex Fragkos, Anastasios Lacey, Michael T. Mena, Dario Reguera, Maria Carmen Number Theory Classical Analysis and ODEs Dynamical Systems Let $\mathcal C= \{k_1<k_2 < \cdots\}$ be Cantor set of integers, that is a set of integers with restricted digits modulo a base $b$, and suppose $0$ is one of the restricted digits. We show that $$ \liminf_N \Expectation_{n\in [N]} m(A\cap T^{-k_n} A \cap \cdots \cap T^{-\ell k_n} A )>0. $$ This is an extension of the IP Ergodic Theorem of Furstenberg and Katznelson, and a partial extension of recent work of Kra and Shalom. In particular, this implies that for any subset of integers $A$ of positive upper Banach density, there is a set $B$ of integers $n$ of positive lower Banach density such that $A$ contains an $\ell+1$ term progression, with step size $k_n$, where $n\in B$. This is a complement to recent results of Kra and Shalom, for IP Sets of integers, and Burgin, concerning Sarkozy's Theorem for Primes with restricted digits. |
| title | Szemerédi's Theorem Along Cantor Sets of Integers |
| topic | Number Theory Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2602.15299 |