On Point Sets in Vector Spaces over Finite Fields That Determine Only Acute Angle Triangles

TL;DR

This paper investigates the maximum size of point sets in finite field vector spaces where all triples form only acute angle triangles, extending classical Euclidean geometric questions to finite fields.

Contribution

It provides an upper bound on the size of such sets in finite fields, offering a new interpretation of acute angles in this algebraic setting.

Findings

Derived an upper bound for the size of sets with only acute triangles in finite fields

Established a finite field analogue of a classical Euclidean geometric problem

Connected finite field geometry with classical Euclidean angle problems

Abstract

For three points $\vec{u}$,$\vec{v}$ and $\vec{w}$ in the $n$-dimensional space $\F_q^n$ over the finite field $\F_q$ of $q$ elements we give a natural interpretation of an acute angle triangle defined by this points. We obtain an upper bound on the size of a set $\cZ$ such that all triples of distinct points $\vec{u}, \vec{v}, \vec{w} \in \cZ$ define acute angle triangles. A similar question in the real space $\cR^n$ dates back to P. Erd{\H o}s and has been studied by several authors.