Fractal transference principle for continued fractions of Laurent series

Abstract

We establish a fractal transference principle for continued fraction expansions over the field of Laurent series. Let $S$ be an infinite subset of the set of all polynomials over a finite field of $q$ elements of positive degree with growth density exponent $\alpha \ge 1$, and let $U \subset S$ be a subset of positive relative upper density. We prove that there exists a subset $E_{S,U}$ of the set of points whose continued fraction digits are pairwise distinct and belong to $S$ such that \[\dim_{\rm H} E_{S,U}=\frac{1}{2\alpha}.\] Moreover, the set of digits appearing in the continued fraction expansions of points in $E_{S,U}$ recovers the relative upper density of $U$ in $S$. We also show that the same construction preserves the relative upper density of the corresponding degree sets in $\mathbb N$. As a consequence, combinatorial statements for subsets of $\mathbb N$ of positive upper density can be transferred to degree sets arising from continued fraction expansions of Laurent series on sets of optimal Hausdorff dimension.