On restricted Falconer distance sets

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 align*Δᵈⁱᵃᵍ(E)= \ \,|(x,x,,x)-(y₁,y₂,,yₖ₋₁)| : x, y₁, ,yₖ₋₁ ∈ E\, * for a compact E⊂ Rᵈ and k≥ 3 . We show that Δ ᵈⁱᵃᵍ(E) has non-empty interior if the Hausdorff dimension of E satisfies (0.1) align (E)> cases 2d+13, & k=3, \\ (k-1)dk,& k≥ 4. cases align We prove an extension of this to C^ω 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.