On restricted Falconer distance sets

Abstract

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.