2004/09/16 by A. Iosevich, Iosevich, A., M. Rudnev +1
Mathematics · #42B #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT) #math.CA #math.NT #msc:42B
paper · pdf · doi:10.48550/arxiv.math/0409283
13 pages
arxiv created 2004/09/16 · arxiv updated 2009/12/01
Let S ⊂ \mathbb Rd be contained in the unit ball. Let Δ(S)=\||a-b||:a,b ∈ S\, the Euclidean distance set of S. Falconer conjectured that the Δ(S) has positive Lebesque measure if the Hausdorff dimension of S is greater than (d)/(2). He also produced an example, based on the integer lattice, showing that the exponent (d)/(2) cannot be improved. In this paper we prove the Falconer distance conjecture for this class of sets based on the integer lattice. In dimensions four and higher we attain the endpoint by proving that the Lebesgue measure of the resulting distance set is still positive if the Hausdorff dimension of S equals (d)/(2). In three dimensions we are off by a logarithm. More generally, we consider K-distance sets ΔK(S)=\|a-b|K: a,b ∈ S\, where |⋅|K is the distance induced by a norm defined by a smooth symmetric convex body K whose boundary has everywhere non-vanishing Gaussian curvature. We prove that our endpoint result still holds in this setting, providing a further illustration of the role of curvature in this class of problems.