On a Continuous Sárközy-Type Problem

We prove that there exists a constant $\epsilon> 0$ with the following property: if $K \subset {\mathbb {R}}^2$ is a compact set that contains no pair of the form $\{x, x + (z, z^{2})\}$ for $z \neq 0$, then $\dim _{\textrm {H}} K \leq 2 - \epsilon $.