Two-point patterns determined by curves

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 Sárk\"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.