On the existence of infinite, non-trivial $F$-sets

TL;DR

This paper proves the existence of infinite, non-trivial $F$-sets over polynomial rings for all finite fields and introduces a new measure called width to analyze their complexity.

Contribution

It confirms a conjecture about the existence of such $F$-sets and refines it by defining and analyzing the width as a complexity measure.

Findings

Existence of infinite, non-trivial $F$-sets over $ ext{F}_q[x]$ for all finite fields.

Introduction of the width concept to measure $F$-set complexity.

Proof of a refined conjecture involving the width of $F$-sets.

Abstract

In this paper we prove a conjecture of J. Andrade, S. J. Miller, K. Pratt and M. Trinh, showing the existence of a non trivial infinite $F$-set over $\mathbb F_q[x]$ for every fixed $q$. We also provide the proof of a refinement of the conjecture, involving the notion of width of an $F$-set, which is a natural number encoding the complexity of the set.