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

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.