Simplices over finite fields

TL;DR

This paper proves that large subsets of finite vector spaces contain isometric copies of certain simplices with self-orthogonal properties, advancing understanding of geometric configurations over finite fields.

Contribution

It establishes new conditions under which simplices with self-orthogonal properties are guaranteed to exist in large finite field subsets.

Findings

Large subsets of f_q^d contain specific self-orthogonal simplices

Conditions on dimensions ensure the existence of these simplices

Results extend previous geometric configurations in finite fields

Abstract

We prove that, provided $d > k$, every sufficiently large subset of $\mathbf{F}_q^d$ contains an isometric copy of every $k$-simplex that avoids spanning a nontrivial self-orthogonal subspace. We obtain comparable results for simplices exhibiting self-orthogonal behavior.