Distribution of simplices in the discrete and continuous settings

2026/08/02 by Thang Pham, Chun-Yen Shen, Boqing Xue
Mathematics · #math.NT #math.CA #math.CO

paper · pdf

44 pages

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

In this paper, we study the distribution of simplices in both discrete and continuous settings. Let q be an odd prime power, let Q be a nondegenerate quadratic form on \mathbb F_q^d, and let 2≤ k≤ d-1. We prove that every set E⊂\mathbb F_q^d with |E|≥ C_d,kq^β_d,k, β_d,k= \begincases (d+k)/(2)-(k-1)/(k+1), d-k even,
(d+k-1)/(2), d-k odd, \endcases determines a positive proportion of all ordered nondegenerate k-simplex congruence classes. This improves the previous exponent due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2017), and is sharp when d-k is odd. In the Euclidean setting, we prove that if E⊂\mathbb R^d is compact and dim_{\mathrm H}(E)>d-1, then there exists a Frostman probability measure μ, supported on E, and a set of pins of full μ-measure such that the pinned distance configuration measure for labeled (d-1)-simplices is absolutely continuous at every such pin. We also show that the same conclusion holds when E⊂\mathbb R^d is a compact Salem set with dim_{\mathrm H}(E)>k.

Citations

• On Erdos-Falconer distance problem in even dimensions
• Triangles with one fixed side-length, a Furstenberg type problem, and incidences in finite vector spaces
• Nonempty interior of configuration sets via microlocal partition optimization
• Pinned simplices and connections to product of sets on paraboloids
• A Mattila-Sjölin theorem for simplices in low dimensions
• Congruence classes of large configurations in vector spaces over finite\n fields
• An elementary approach to simplexes in thin subsets of Euclidean space
• Simplices over finite fields
• Finite configurations in sparse sets
• Multi-parameter projection theorems with applications to sums-products and finite point configurations in the Euclidean setting
• Extension and averaging operators for finite fields
• The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs
• Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates
• On the solvability of systems of bilinear equations in finite fields
• On kaleidoscopic pseudo-randomness of finite Euclidean graphs