2016/10/02 by Alex Iosevich, Iosevich, Alex, Krystal Taylor +3
Mathematics · #28A75 #42B10 #53C10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #math.CA #math.MG #msc:28A75 #msc:42B10 #msc:53C10
paper · pdf · doi:10.48550/arxiv.1610.00349
arxiv created 2016/10/02 · arxiv updated 2016/10/04
Let M be a compact d-dimensional Riemannian manifold without a boundary. Given E ⊂ M, let Δρ(E)=\ρ(x,y): x,y ∈ E \, where ρ is the Riemannian metric on M. Let Δρx denote the pinned distance set, namely, \ρ(x,y): y ∈ E \ with x ∈ E. We prove that if the Hausdorff dimension of E is greater than (d+1)/(2), then there exist many x ∈ E such that the Lebesgue measure of Δxρ(E) is positive. This result was previously established by Peres and Schlag in the Euclidean setting. The main result is deduced from a variable coefficient Euclidean formulation, which can be used to study a variety of geometric problems. We extend our result to the setting of chains studied in \citeBIT15 and obtain a pinned estimate in this context. Moreover, we point out that our scheme is quite universal in nature and this idea will be exploited in variety of settings in the sequel.