Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 579-585

FALCONER DISTANCE PROBLEM, ADDITIVE ENERGY AND CARTESIAN PRODUCTS

Alex Iosevich and Bochen Liu

University of Rochester, Department of Mathematics
RC Box 270138, Rochester, NY 14627, U.S.A.; iosevich 'at' math.rochester.edu

University of Rochester, Department of Mathematics
RC Box 270138, Rochester, NY 14627, U.S.A.; bochen.liu 'at' rochester.edu

Abstract. A celebrated result due to Wolff says 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 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 d²/2d–1. The proof uses a combination of Fourier analysis and additive comibinatorics.

2010 Mathematics Subject Classification: Primary 28A75, 52C10.

Key words: Distance problem, Cartesian products, additive energy, Ahlfors-David regular.

Reference to this article: A. Iosevich and B. Liu: Falconer distance problem, additive energy and Cartesian products. Ann. Acad. Sci. Fenn. Math. 41 (2016), 579-585.

Full document as PDF file

doi:10.5186/aasfm.2016.4135