  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.
