Congruence Classes of Simplex Structures in Finite Field Vector Spaces

Abstract

We study a generalization of the Erd\H{o}s-Falconer distance problem over finite fields. For a graph $G$, two embeddings $p, p': V(G) \to \mathbb{F}_q^d$ of a graph $G$ are congruent if for all edges $(v_i, v_j)$ of $G$ we have that $||p(v_i) - p(v_j)|| = ||p'(v_i) - p'(v_j)||$. What is the infimum of $s$ such that for any subset $E\subset \mathbb{F}_q^d$ with $|E| \gtrsim q^s$, $E$ contains a positive proportion of congruence classes of $G$ in $\mathbb{F}_q^d$? Bennett et al. and McDonald used group action methods to prove results in the case of $k$-simplices. The work of Iosevich, Jardine, and McDonald as well as that of Bright et al. have proved results in the case of trees and trees of simplices, utilizing the inductive nature of these graphs. Recently, Aksoy, Iosevich, and McDonald combined these two approaches to obtain nontrivial bounds on the "bowtie" graph, two triangles joined at a vertex. Their proof relies on an application of the Hadamard three-lines theorem to pass to a different graph. We develop novel geometric techniques called branch shifting and simplex unbalancing to reduce our analysis of trees of simplices to a much smaller class of simplex structures. This allows us to establish a framework that handles a wide class of graphs exhibiting a combination of rigid and loose behavior. In $\mathbb{F}_q^2$, this approach gives new nontrivial bounds on chains and trees of simplices. In $\mathbb{F}_q^d$, we improve on the results of Bright et al. in many cases and generalize their work to a wider class of simplex trees. We discuss partial progress on how this framework can be extended to more general simplex structures, such as cycles of simplices and structures of simplices glued together along an edge or a face.