The Cauchy-Schwarz inequality in Cayley graph and tournament structures on finite fields

TL;DR

This paper explores analogues of the Cauchy-Schwarz inequality within Cayley graph and tournament structures on finite fields, revealing conditions under which these inequalities hold in finite vector spaces.

Contribution

It introduces finite field structures where the Cauchy-Schwarz inequality analogues are valid, extending classical inequalities to new algebraic and combinatorial contexts.

Findings

Cayley graph structures induce a Manhattan norm on finite vector spaces.

An analogue of the Cauchy-Schwarz inequality holds with respect to the Manhattan norm.

The inequality is valid in large neighborhoods around the null vector for large primes.

Abstract

The Cayley graph construction provides a natural grid structure on a finite vector space over a field of prime or prime square cardinality, where the characteristic is congruent to 3 modulo 4, in addition to the quadratic residue tournament structure on the prime subfield. Distance from the null vector in the grid graph defines a Manhattan norm. The Hermitian inner product on these spaces over finite fields behaves in some respects similarly to the real and complex case. An analogue of the Cauchy-Schwarz inequality is valid with respect to the Manhattan norm. With respect to the non-transitive order provided by the quadratic residue tournament, an analogue of the Cauchy-Schwarz inequality holds in arbitrarily large neighborhoods of the null vector, when the characteristic is an appropriate large prime.