Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 713-720

SHARPNESS OF FALCONER'S (d+1)/2 ESTIMATE

Alex Iosevich and Steven Senger

University of Rochester, Mathematics Department
Rochester, NY 14627, U.S.A.; alex.iosevich 'at' rochester.edu

Missouri State University, Department of Mathematics
Springfield, MO 65897, U.S.A.; stevensenger 'at' missouristate.edu

Abstract. In the paper introducing the celebrated Falconer distance problem, Falconer proved that the Lebesgue measure of the distance set is positive, provided that the Hausdorff dimension of the underlying set is greater than (d+1)/2. His result is based on the estimate

(0.1) μ × μ {(x,y) : 1 ≤ |x – y| ≤ 1 + ε} ∼≤ ε,

where μ is a Borel measure satisfying the energy estimate I_s(μ) = ∫∫ |x – y|^-s dμ(x) dμ(y) < ∞ for s > (d+1)/2. An example due to Mattila [12, Remark 4.5], [11] shows in two dimensions that for no s < 3/2 does I_s(μ) < ∞ imply (0.1). His construction can be extended to three dimensions. Mattila's example readily applies to the case when the Euclidean norm in (0.1) is replaced by a norm generated by a convex body with a smooth boundary and non-vanishing Gaussian curvature. In this paper we prove, for all d ≥ 2, that for no s < (d+1)/2 does I_s(μ) < ∞ imply (0.1) or the analogous estimate where the Euclidean norm is replaced by the norm generated by a particular convex body B with a smooth boundary and everywhere non-vanishing curvature. Our construction is based on a combinatorial construction due to Valtr [15].

2010 Mathematics Subject Classification: Primary 28A75, 51A20, 52C10.

Key words: Geometric measure theory, discrete geometric combinatorics.

Reference to this article: A. Iosevich and S. Senger: Sharpness of Falconer's (d+1)/2 estimate. Ann. Acad. Sci. Fenn. Math. 41 (2016), 713-720.

Full document as PDF file

doi:10.5186/aasfm.2016.4145