Falconer distance problem, additive energy and Cartesian products

Alex Iosevich, Bochen Liu
2016 Annales Academiae Scientiarum Fennicae: Mathematica  
A celebrated result due to Wolff says if E is a compact subset of R 2 , then the Lebesgue measure of the distance set ∆(E) = {|x − y| : x, y ∈ E} is positive if the Hausdorff dimension 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 d 2 2d−1 . The proof uses a combination of Fourier analysis and additive comibinatorics.
doi:10.5186/aasfm.2016.4135 fatcat:hnbo3lhwh5cufnap3mbfb7zq6i