7 papers
On the proof of Falconer distance conjecture
Kotaro Inami, Jianbin Li, Yudong Liang +2
We are able to catch the main idea from OpenAI's recent release and present a proof of Falconer distance conjecture in our own words. Arguments are judiciously simplified and caref…
On volume vectors determined by hypergraphs in thin subsets of Euclidean space
Tainara Borges, Ben Foster, Yumeng Ou +2
Generalizing the Falconer distance problem, the authors of this paper recently established the first non-trivial dimensional threshold for any distance graph in high enough of a di…
Falconer-type results for any finite graph with multiple pins
Tainara Borges, Ben Foster, Yumeng Ou +2
A generalization of the celebrated Falconer distance problem asks for a graph , with vertex set and edge set , how large the…
Quantifying Extrinsic Curvature in Neural Manifolds
Francisco Acosta, Sophia Sanborn, Khanh Dao Duc +2
The neural manifold hypothesis postulates that the activity of a neural population forms a low-dimensional manifold whose structure reflects that of the encoded task variables. In…
VC-Dimension and Distance Chains in
Ruben Ascoli, Livia Betti, Justin Cheigh +8
Given a domain and a collection of functions , the Vapnik-Chervonenkis (VC) dimension of measures its complexity in an appropriate s…
Distinct Angles and Angle Chains in Three Dimensions
Ruben Ascoli, Livia Betti, Jacob Lehmann Duke +6
In 1946, Erdős posed the distinct distance problem, which seeks to find the minimum number of distinct distances between pairs of points selected from any configuration of poin…
A Mattila-Sjölin theorem for triangles
Eyvindur Ari Palsson, Francisco Romero Acosta
We show for a compact set , , that if the Hausdorff dimension of is larger than , then the set of congruence classes of triang…