Pinned Geometric Configurations in Euclidean Space and Riemannian Manifolds - Alex Iosevich; Krystal Taylor; Ignacio Uriarte-Tuero (Mathematics, 2021)
Let M be a compact d-dimensional Riemannian manifold without a boundary. Given a compact set <semantics>E⊂M</semantics>, we study the set of distances from the set E to a fixed point <semantics>x∈E</semantics>. This set is <semantics><msubsup>Δρx</msubsup>(E)={ρ(x,y):y∈E}</semantics>, where <semantics>ρ</semantics> is the Riemannian metric on M. We prove that if the Hausdorff dimension of E is greater than <semantics>d+12</semantics>, then there exist many <semantics>x∈E</semantics> such that the Lebesgue measure of <semantics><msubsup>Δρx</msubsup>(E)</semantics> 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.