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 Sárközy-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 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 \({\mathbb {R}}^d\) with 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.
