Sanju Velani: Intersecting well approximable and missing digit sets

Abstract: Let \(b \geq 3\) be an integer and \(C(b,D)\) be the set of real numbers whose \(b\)-ary expansion consists of digits restricted to a given set \(D \subseteq \{0,\ldots,b-1\}\). Given an integer \(t \geq 2\) and a real, positive function \(\psi\), let \(W_t(\psi)\) denote the set of real numbers \(x\) for which \(|x-p/t^n| < \psi(n)\) for infinitely many \((p,n) \in \mathbb{Z} \times \mathbb{N}\). In this talk we discuss recent results concerning the Hausdorff measure and dimension of the intersection of \(W_t(\psi)\) and \(C(b,D)\). This line of work is motivated by a question posed by Mahler in the 1980s: how close can irrational elements of the middle third Cantor set be approximated by rational numbers?