Suppose $(X,d)$ is a metric space equipped with a Borel probability measure and suppose $(B_i)_{i \in \mathbb{N}}$ is a sequence of measurable sets in $X$. Suppose $T: X \to X$ is a measure preserving transformation and consider the set: \[\mathcal{B} = \{x \in X: T^n x \in B_n \text{ for infinitely many } n\in\mathbb{N}\}. \] This is a shrinking target set. The terminology "shrinking targets" was first introduced by Hill and Velani in 1995. Since then, shrinking target problems have received a great deal of interest, especially with regards to studying the measure-theoretic and dimension-theoretic properties of shrinking target sets. In this talk, I will discuss some recent work with Thomas Jordan (Bristol, UK) and Ben Ward (York, UK) where we establish the Hausdorff dimension of a shrinking target set where our "targets" (the $B_n$) are rectangles and $X$ is a self-similar carpet.
