Almost orthogonal subsets of vector spaces over finite fields

TL;DR

This paper investigates the size and structure of subsets in finite field vector spaces that contain many mutually orthogonal pairs, providing new bounds and variants of classical theorems.

Contribution

It introduces sharp finite field variants of Rosenfeld's theorem and an approximate version of Berlekamp's theorem, advancing understanding of orthogonal subsets.

Findings

Derived bounds on the size of orthogonal subsets

Established finite field analogs of classical theorems

Provided structural insights into orthogonal vector configurations

Abstract

We prove various results on the size and structure of subsets of vector spaces over finite fields which, in some sense, have too many mutually orthogonal pairs of vectors. In particular, we obtain sharp finite field variants of a theorem of Rosenfeld and an almost version of a theorem of Berlekamp.