1. Pinned Geometric Configurations in Euclidean Space and Riemannian Manifolds
- Author
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Alex Iosevich, Krystal Taylor, and Ignacio Uriarte-Tuero
- Subjects
pinned distance sets ,Falconer conjecture ,Riemannian manifolds ,fractals ,Mathematics ,QA1-939 - Abstract
Let M be a compact d-dimensional Riemannian manifold without a boundary. Given a compact set E⊂M, we study the set of distances from the set E to a fixed point x∈E. This set is Δρx(E)={ρ(x,y):y∈E}, where ρ is the Riemannian metric on M. We prove that if the Hausdorff dimension of E is greater than d+12, then there exist many x∈E such that the Lebesgue measure of Δρx(E) is positive. This result was previously established by Peres and Schlag in the Euclidean setting. We give a simple proof of the Peres–Schlag result and generalize it to a wide range of distance type functions. Moreover, we extend our result to the setting of chains studied in our previous work and obtain a pinned estimate in this context.
- Published
- 2021
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