A quantitative form of the planar Falconer distance theorem

We give a quantitative planar form of a multiscale argument for Falconer's distance problem. For every $0<η<1$, we obtain a positive lower bound for the Lebesgue measure of the distance set of a compact subset of the unit ball supporting a probability measure with $ν(B(x,r))\le 2r^{1+η}$. The bound