An $L^2$ identity and pinned distance problem

Given a measure on a subset of Euclidean spaces. The $L^2$ spherical averages of the Fourier transform of this measure was originally used to attack Falconer distance conjecture, via Mattila’s integral. In this talk, we will consider pinned distance problem, a stronger version of Falconer distance problem, and show that spherical averages imply the same dimensional threshold on both problems. In particular, with the best known spherical averaging estimates, we improve a result of Peres and Schlag on pinned distance problem significantly. The idea is to reduce the pinned distance problem to an integral where spherical averages apply. The key new ingredient is an identity between square functions.