Sums of Laurent series with bounded partial quotients

TL;DR

The paper develops Laurent-series analogues of classical real-number results on sums of elements with constrained partial quotients, introducing $\mathcal{F}(k)$, $\mathcal{S}(k)$, and $\mathcal{G}$ for $\mathbb{K}((t^{-1}))$ and proving Hall- and Shulga-type results in this setting. It shows that, for $\mathbb{K}\neq\mathbb{F}_2$, every element of $\mathbb{K}((t^{-1}))$ can be written as a sum of two elements from $\mathcal{F}(1)$, with a corresponding result $\mathcal{F}(2)+\mathcal{F}(1)$ when $\mathbb{K}=\mathbb{F}_2$, and establishes analogous sumset properties for $\mathcal{S}(k)$ and $\mathcal{G}$ (including a stronger $\mathcal{G}'$ variant). It provides a constructive Laurent-series version of Shulga’s algorithm yielding $\alpha=\beta+\gamma$ with controlled growth of partial quotients and a finite termination bound for rational inputs, highlighting both the general theory and the characteristic-two nuances. Overall, the work extends real-analytic Diophantine ideas to the function-field context, offering explicit decomposition methods via Hankel-determinant techniques and continued-fraction analysis.

Abstract

In 1947 M.Hall proved that every real number is the sum of an integer and two real numbers whose partial quotients are at most $4$. Later, Cusick proved that every real number is the sum of an integer and two real numbers whose partial quotients are at least $2$. In a recent paper, the authors proved that every real number is the sum of two real numbers whose partial quotients diverge. In this paper, we prove an analogue of these results for Laurent series.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Definition 6: Shulga
• Theorem 7: shulga and GN
• Definition 8
• Theorem 9