Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets

Let $d\ge 3$, $$D⊊\{0,1,\dots ,d-1\},\mathrm{\mid }D\mathrm{\mid }\ge 2,0\in D$$ be a finite alphabet, and define the integer Cantor set \begin{align} $\mathcal{C}$ := $\mathcal{C}${D} := \bigcup{J \geq 0} \Big{ \sum_{j =0}^J a_j d^j : a_j \in D \Big}. \end{align} We prove that for any $\sigma$-finite measure-preserving system, $(X,\mu ,T)$, and any $f\in L^p(X)$, $2\le p<\mathrm{\infty }$, the ergodic averages \begin{align} \frac{1}{|$\mathcal{C}$N|} \sum{n \in $\mathcal{C}$N } f(T^n x), \qquad $\mathcal{C}$N := $\mathcal{C}$ \cap {1,2,\dots,N } \end{align} converge $\mu$-almost everywhere. By rescaling, this allows us to resolve the question of lacunary differentiation of Cantor measures at self-similar scales: if \begin{align} $\mathcal{C}$’ := \Big{ \sum{j \geq 1} a_j d^{-j} : a_j \in D \Big} \subset [0,1] \end{align} is a real-variable Cantor set, and $\nu$ denotes its natural measure, then we prove that \begin{align} \lim{k \to \infty} \int f(x-d^{-k} t) \ dν(t) = f(x) \end{align} Lebesgue almost-everywhere for any $f\in L_{\text{loc}}^2(\mathbb{R})$.