Falconer distance problem, additive energy and Cartesian products

A celebrated result due to Wolffsays if E is a compact subset of R², then the Lebesgue measure of the distance set Δ(E) = (|x - y|: x, y ∈ E) is positive if the Hausdorffdimension of E is greater than 4/3. In this paper we improve the 4/3 barrier by a small exponent for Cartesian products. In higher dimensions, also in the context of Cartesian products, we reduce Erdogan's d/2 + 1/3 exponent to d²/2d-1. The proof uses a combination of Fourier analysis and additive comibinatorics.