Improvement on 2-chains inside thin subsets of Euclidean spaces

2017/09/20 by Bochen Liu, Liu, Bochen
Mathematics · #28A75 #42B20 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #math.CA #math.CO #math.MG #msc:28A75 #msc:42B20

paper · pdf · doi:10.48550/arxiv.1709.06814

Compared with the last version, we rewrite the proof in terms of weighted spherical averaging operators. Also we delete the discussion about product of distances since it is not very close to the main topic of this paper

arxiv created 2017/10/25 · arxiv updated 2017/10/26

Abstract

We prove that if the Hausdorff dimension of E⊂ℝ^d, d≥ 2 is greater than (d)/(2)+(1)/(3), the set of gaps of 2-chains inside E, Δ₂(E)=\(|x-y|, |y-z|): x, y, z∈ E \⊂ℝ² has positive Lebesgue measure. It generalizes Wolff-Erdogan's result on distances and improves a result of Bennett, Iosevich and Taylor on finite chains. We also consider the similarity class of 2-chains, S₂(E)=\(t₁)/(t₂):(t₁,t₂)∈Δ₂(E)\=\(|x-y|)/(|y-z|): x, y, z∈ E \⊂ℝ, and show that |S₂(E)|>0 whenever dim_H(E)>(d)/(2)+(1)/(7).

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