English

Falconer distance problem on Riemannian manifolds

Classical Analysis and ODEs 2024-10-24 v3 Analysis of PDEs

Abstract

We prove that on a dd-dimensional Riemannian manifold, the distance set of a Borel set EE has a positive Lebesgue measure if dimH(E)>d2+14+1(1)d8d.\dim_{\mathcal{H}}(E)>\frac d2+\frac14+\frac{1-(-1)^d}{8d}.

Keywords

Cite

@article{arxiv.2309.01204,
  title  = {Falconer distance problem on Riemannian manifolds},
  author = {Changbiao Jian and Bochen Liu and Yakun Xi},
  journal= {arXiv preprint arXiv:2309.01204},
  year   = {2024}
}

Comments

18 pages.We have removed the constant sectional curvature condition, which has led to a strengthened version of the main result