On a continuous Sárközy type problem

Abstract

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

18 pages. v4: Added references