Two-point patterns determined by curves

Let Γ⊂ Rᵈ be a smooth curve containing the origin. Does every Borel subset of Rᵈ of sufficiently small codimension enjoy a Sárközy-like property with respect to $Γ$, namely, contain two elements differing by a member of Γ \0\? Kuca, Orponen, and Sahlsten have answered this question in the affirmative for a specific curve with nonvanishing curvature, the standard parabola (t, t²) in R². 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 Γ⊂ Rᵈ of finite type at the origin, we prove the existence of a dimensional threshold ε >0 such that every Borel subset of Rᵈ of Hausdorff dimension larger than d - ε contains a pair of points of the form , x+γ\ with γ∈ Γ \0\. The threshold ε 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 $Γ$ is necessary, provided $Γ$ 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.