Different bases in investigation of $\sqrt[3]{2}$

TL;DR

This paper investigates the continued fraction expansion of b2b2, b2b3, using various bases in a vector space to analyze the properties of partial quotients and their boundedness.

Contribution

It introduces a new approach using different bases in a vector space to establish criteria for continued fraction convergents of b2b2.

Findings

Numerical experiments support boundedness conjectures.

A criterion for continued fraction convergents based on lattice coefficient vectors.

Application of multiple bases provides new insights into the structure of b2b2's continued fractions.

Abstract

The present paper is in a sense a continuation of \cite{PLS}, it relies on the notation and some results. The problem tackled in both papers is the nature of the continued fraction expansion of $\sqrt[3]{2}$: are the partial quotients bounded or not. Numerical experiments suggest an even stronger result on the lines of Kuzmin statistics. Here we apply different sets of bases for the vector space $V$, where the adjunction ring $\ZZ\lbrack \sqrt[3]{2} \rbrack$ lives. And as a result we get a criterion for continued fraction convergents in terms of the coefficient vector from a lattice.