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

TL;DR

This paper extends Szemerédi's Theorem to Cantor sets of integers with restricted digits, showing that certain multiple recurrence properties hold along these sets, implying the existence of arithmetic progressions with steps in these sets.

Contribution

It generalizes the IP Ergodic Theorem to Cantor sets of integers and establishes new recurrence results for subsets of integers with positive density.

Findings

Positive lower Banach density sets contain progressions with steps in Cantor sets.

Extension of Furstenberg-Katznelson IP Ergodic Theorem to restricted digit sets.

Partial extension of Kra and Shalom's work on IP sets.

Abstract

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.