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The Cayley graph construction provides a natural grid structure on a finitevector space over a field of prime or prime square cardinality, where thecharacteristic is congruent to 3 modulo 4, in addition to the quadratic residuetournament structure on the prime subfield. Distance from the null vector inthe grid graph defines a Manhattan norm. The Hermitian inner product on thesespaces over finite fields behaves in some respects similarly to the real andcomplex case. An analogue of the Cauchy-Schwarz inequality is valid withrespect to the Manhattan norm. With respect to the non-transitive orderprovided by the quadratic residue tournament, an analogue of the Cauchy-Schwarzinequality holds in arbitrarily large neighborhoods of the null vector, whenthe characteristic is an appropriate large prime.

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