On a continuous S\'ark\"ozy type problem

TL;DR

This paper proves that compact sets in the plane avoiding certain quadratic patterns have Hausdorff dimension strictly less than 2, establishing a quantitative restriction on their size.

Contribution

It introduces a new bound on the Hausdorff dimension of sets avoiding quadratic difference patterns, extending Sárközy-type problems to continuous geometric settings.

Findings

Sets avoiding quadratic patterns have Hausdorff dimension at most 2 - ε.

Existence of a positive ε such that the dimension bound holds.

Provides a quantitative measure of the size of pattern-avoiding sets.

Abstract

We prove that there exists a constant $\varepsilon > 0$ 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 $z \neq 0$, then $\mathrm{dim}_\mathrm{H} K \leq 2 - \varepsilon$.