Given a decreasing family $B_1\supset B_2 \supset\dotsb$ of targets in a measure space $X$ equipped with a flow $\varphi_t$ (or transformation), the shrinking target problem asks to characterize when there is a full measure set of points $x$ that hit the targets infinitely often in the sense that $\{n \in \mathbb{N} \mid \varphi_n(x)\in B_n\}$ is unbounded.

This talk will examine the discrete shrinking target problem for the Teichmüller flow on the moduli space of unit-area quadratic differentials. Specifically, for any nested sequence of measurable sets $B_i$ in the moduli space of Riemann surfaces with preimages $E_i$ in quadratic differential space, consider the set $\mathcal{H}$ of unit-area quadratic differentials that hit the targets $E_i$ infinitely often. We show that for any $\mathrm{SL}(2,\mathbb{R})$--invariant probability measure $\mu$, the set $\mathcal{H}$ has zero measure if $\{\mu(E_i)\}$ is summable and otherwise has full measure. As an application, we obtain logarithm laws describing how quickly a generic trajectory $\{\varphi_n(q) \mid n\in \mathbb{N}\}$ accumulates on a given point. Joint with Grace Work.