  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.
