In this talk, I will discuss joint work with Malabika Pramanik on the problem of locating patterns in sets of high Hausdorff dimension. More specifically, suppose $\Gamma$ is a smooth curve in Euclidean space that passes through the origin. Is it true that every set with sufficiently high Hausdorff dimension must contain two distinct points $x,y$ such that $x-y \in \Gamma$? We showed that if $\Gamma$ is suitably curved then the answer is yes, while for certain flat curves the answer is no. This generalizes work of Kuca, Orponen, and Sahlsten, who answered this question affirmatively when $\Gamma$ is the standard parabola in $\mathbb{R}^2$.