Fractal transference principle for continued fractions of Laurent series

TL;DR

This paper develops a fractal transference principle for continued fraction expansions over Laurent series fields, linking Hausdorff dimension and density properties of digit sets.

Contribution

It introduces a novel method to transfer combinatorial properties from subsets of natural numbers to digit sets in Laurent series continued fractions.

Findings

Constructed sets with Hausdorff dimension 1/(2α) for digit sets with density exponent α ≥ 1.

Showed that the density of digit sets in Laurent series expansions reflects the density of subsets in natural numbers.

Established a link between Hausdorff dimension and combinatorial density transfer in Laurent series continued fractions.

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.