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 $$ \begin{align*}\Delta^{\mathrm{diag}}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}\end{align*} $$for a compact $E\subset \mathbb {R}^d$ and $k\ge 3$. We show that $\Delta ^{\mathrm{diag}}(E)$ has non-empty interior if the Hausdorff dimension of E satisfies (0.1)$$ \begin{align} \dim(E)> \begin{cases} \frac{2d+1}3, & k=3, \\ \frac{(k-1)d}k,& k\ge 4. \end{cases} \end{align} $$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.