F-sets of arbitrary finite width

Ferraguti and Micheli introduced the width of an $F$-set and conjectured that non-trivial $F$-sets of arbitrary width exist over every finite field. For $q\mathrm{\ne }2,3$, their constructions give examples of widths one and two. We prove that, for every $q\mathrm{\ne }2,3$ and every integer $r\ge 1$, there exists an infinite, non-trivial $F$-set in ${\mathbb{F}}_q[X]$ of width exactly $r$. Thus the finite-width part of their conjecture is settled over all such fields. The proof combines a bounded-core family of irreducible power substitutions with factorization results for $g(X^n)$, Dirichlet’s theorem over ${\mathbb{F}}_q[X]$, and Kummer lifting. Core degree gives a uniform upper bound on the width, while parallel successor ladders give the required lower bound; a suitable tail of the nullity filtration then has the prescribed width.