Pinned distance problem, slicing measures, and local smoothing estimates

We improve on the Peres–Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with \[ Δ y ( E ) = { | x − y | : x ∈ E } , Δ ^y(E) = \{|x-y|:x∈ E\}, \] we prove that for any E , F ⊂ R d E, F⊂ {R}^d , there exists a probability measure μ F μ _F on F F such that for μ F μ _F -a.e. y ∈ F y∈ F , dim H ( Δ y ( E ) ) ≥ β { _{{ H}}}(Δ ^y(E))≥ β if dim H