Two-point patterns determined by curves

Abstract

Let $\Gamma \subset \mathbb{R}^d$ be a smooth curve containing the origin. Does every Borel subset of $\mathbb R^d$ of sufficiently small codimension enjoy a S\'ark\"ozy-like property with respect to $\Gamma$, namely, contain two elements differing by a member of $\Gamma \setminus \{0\}$? Kuca, Orponen, and Sahlsten have answered this question in the affirmative for a specific curve with nonvanishing curvature, the standard parabola $(t, t^2)$ in $\mathbb{R}^2$. In this article, we use the analytic notion of "functional type", a generalization of curvature ubiquitous in harmonic analysis, to study containment of patterns in sets of large Hausdorff dimension. Specifically, for $\textit{every}$ curve $\Gamma \subset \mathbb{R}^d$ of finite type at the origin, we prove the existence of a dimensional threshold $\varepsilon >0$ such that every Borel subset of $\mathbb{R}^d$ of Hausdorff dimension larger than $d - \varepsilon$ contains a pair of points of the form $\{x, x+\gamma\}$ with $\gamma \in \Gamma \setminus \{0\}$. The threshold $\varepsilon$ we obtain, though not optimal, is shown to be uniform over all curves of a given "type". We also demonstrate that the finite type hypothesis on $\Gamma$ is necessary, provided $\Gamma$ either is parametrized by polynomials or is the graph of a smooth function. Our results therefore suggest a correspondence between sets of prescribed Hausdorff dimension and the "types" of two-point patterns that must be contained therein.