Szemerédi's Theorem Along Cantor Sets of Integers

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 $$\underset{N}{\mathrm{lim\hspace{0.17em}inf}}{\text{Expectation}}_{n\in [N]}m(A\cap T^{-k_n}A\cap \cdots \cap T^{-\mathrm{\ell }{k}_n}A)>0.$$ 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 $\mathrm{\ell }+1$ 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.