  We introduce a class of Falconer distance problems, which we call of
restricted type, lying between the classical version and its pinned variant.
Prototypical restricted distance sets are the diagonal distance sets, $k$-point
configuration sets given by $$\Delta^{diag}(E)= \{
\,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}$$
for a compact $E\subset\mathbb{R}^d$ and $k\ge 3$. We show that
$\Delta^{diag}(E)$ has non-empty interior if the Hausdorff dimension of $E$
satisfies \begin{equation*} \dim(E) > \begin{cases} \frac{2d+1}3, & k=3 \\
\frac{(k-1)d}k,& k\ge 4. \end{cases} \end{equation*} We prove an extension of
this to $C^\omega$ Riemannian metrics $g$ close to the product of Euclidean
metrics. For product metrics this follows from known results on pinned distance
sets, but to obtain a result for general perturbations $g$ we present a
sequence of proofs of partial results, leading up to the proof of the full
result, which is based on estimates for multilinear Fourier integral operators.
