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arXiv · 2602.15299

Szemer\'edi's Theorem Along Cantor Sets of Integers

Alex Burgin·Anastasios Fragkos·Michael T. Lacey·Dario Mena·Maria Carmen Reguera
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Abstract

Let $\mathcal C= \{k_1 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.

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Alex Burgin, Anastasios Fragkos, Michael T. Lacey, Dario Mena, Maria Carmen Reguera. 2026-02-17. Szemer\'edi's Theorem Along Cantor Sets of Integers. https://arxiv.org/abs/2602.15299

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