Falconer's distance set problem via the wave equation

Abstract: Falconer proved that there are sets $E\subset \mathbb{R}^n$ of Hausdorff dimension $n/2$ whose distance sets ${|x-y| : x,y\in E}$ are null with respect to Lebesgue measure. This led to the conjecture that distance sets have positive Lebesgue measure as soon the Hausdorff dimension of $E$ is larger than $n/2$. The best results in this direction have exploited estimates that restrict the Fourier transform of measures to the $(n-1)$-dimensional sphere. Here we show that these estimates can be replaced by estimates that restrict the Fourier transform of measures to the $n$-dimensional cone. Such estimates were first considered by Wolff in their adjoint form whereby they bound the solution to the wave equation in terms of its initial data. The connection with Falconer's problem, combined with Falconer's counterexample, provides a new necessary condition for what was considered a plausible conjecture for these estimates.