Extensions of the Carlitz-McConnel and Blokhuis-Sziklai theorems for unions of cyclotomic classes

TL;DR

This paper extends classical theorems about functions with specific difference properties in finite fields to unions of cyclotomic classes and explores related maximum clique problems in Cayley graphs.

Contribution

It generalizes the Carlitz-McConnel theorem to unions of cyclotomic classes and strengthens results on maximum cliques in related Cayley graphs.

Findings

Extended the Carlitz-McConnel theorem to unions of cosets of subgroups.

Strengthened results on maximum cliques in Cayley graphs over finite fields.

Provided new bounds and structural insights for these combinatorial objects.

Abstract

Let $p$ be a prime, let $q=p^n$, and let $D\subseteq \mathbb{F}_q^\ast$. A celebrated result of Carlitz and McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^\ast$, and $f:\mathbb{F}_q\to\mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y)\in D$ for all $x\neq y$, then $f$ must be of the form $f(x)=ax^{p^j}+b$. In this paper, we extend their result to the setting where $D$ is a union of cosets of a fixed subgroup of $\mathbb{F}_q^\ast$, under a mild assumption. In a similar spirit, we also investigate maximum cliques in related Cayley graphs over finite fields, strengthening several results of Blokhuis, Sziklai, and Asgarli and Yip.