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 Delta(diag)(E) = {vertical bar(x, x, ... , x) - (y(1), y(2), ... , y(k-1))vertical bar : x, y(1), ... , y(k-1) is an element of E} for a compact E subset of R-d and k >= 3. We show that Delta(diag)(E) has non-empty interior if the Hausdorff dimension of E satisfies (0.1) dim(E) > {2d+1/3, k = 3, (k-1)d/k, k >= 4. 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.