Sums of Laurent series with bounded partial quotients

TL;DR

This paper extends classical results about sums of real numbers with bounded partial quotients to Laurent series, demonstrating analogous properties in this broader context.

Contribution

It establishes new results for Laurent series, showing that every Laurent series can be expressed as a sum of two with diverging partial quotients, paralleling real number cases.

Findings

Analogous results for Laurent series are proved.

Every Laurent series can be written as a sum of two with diverging partial quotients.

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.