  In this article, we study two problems concerning the size of the set of
finite point configurations generated by a compact set $E\subset \mathbb{R}^d$.
The first problem concerns how the Lebesgue measure or the Hausdorff dimension
of the finite point configuration set depends on that of $E$. In particular, we
show that if a planar set has dimension exceeding $\frac{5}{4}$, then there
exists a point $x\in E$ so that for each integer $k\geq2$, the set of
"$k$-chains" with initial point at $x$ has positive Lebesgue measure.
  The second problem is a continuous analogue of the Erd\H{o}s unit distance
problem, which aims to determine the maximum number of times a point
configuration with prescribed gaps can appear in $E$. For instance, given a
triangle with prescribed sides and given a sufficiently regular planar set $E$
with Hausdorff dimension no less than $\frac{7}{4}$, we show that the dimension
of the set of vertices in $E$ forming said triangle does not exceed
$3\,\dim_{\mathcal{H}} (E)-3$. In addition to the Euclidean norm, we consider
more general distances given by functions satisfying the so-called Phong-Stein
rotational curvature condition. We also explore a number of examples to
demonstrate the extent to which our results are sharp.
