2024/10/23 by Shuai Huang, Huang, Shuai, J. S. Miller +5
Mathematics · #52a35 #52b35 #52c35 #55s91 #Advanced Combinatorial Mathematics #Advanced Differential Equations and Dynamical Systems #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2410.18331
openalex publication_date 2024/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Equipartition theory, beginning with the classical ham sandwich theorem, seeks the fair division of finite point sets in ℝd by the full-dimensional regions determined by a prescribed geometric dissection of ℝd. Here we examine equidistributions of finite point sets in ℝd by prescribed low dimensional subsets. Our main result states that if r≥ 3 is a prime power, then for any m-coloring of a sufficiently small point set X in ℝd, there exists an r-fan in ℝd -- that is, the union of r ``half-flats'' of codimension r-2 centered about a common (r-1)-codimensional affine subspace -- which captures all the points of X in such a way that each half-flat contains at most an r-th of the points from each color class. The number of points in ℝd we require for this is essentially tight when m≥ 2. Additionally, we extend our equidistribution results to ''piercing'' distributions in a similar fashion to Dolnikov's hyperplane transversal generalization of the ham sandwich theorem. By analogy with recent work of Frick et al., our results are obtained by applying Gale duality to linear cases of topological Tverberg-type theorems. Finally, we extend our distribution results to multiple r-fans after establishing a multiple intersection version of a topological Tverberg-type theorem due to Sarkaria.