Constructing Simultaneous Diophantine Approximations of Certain Cubic Numbers

TL;DR

This paper develops a method to construct sequences that approximate certain cubic algebraic numbers simultaneously, providing an effective proof of Littlewood's conjecture for specific pairs using elementary algebraic number theory and continued fractions.

Contribution

It introduces a new constructive approach to simultaneous Diophantine approximation in cubic fields, leading to an effective proof of Littlewood's conjecture for particular algebraic pairs.

Findings

Constructs sequences with specific approximation properties

Provides an elementary proof of Littlewood's conjecture for certain pairs

Establishes bounds on approximation errors using algebraic number theory

Abstract

For $K$ a cubic field with only one real embedding and $\alpha,\beta\in K$, we show how to construct an increasing sequence $\{m_n\}$ of positive integers and a subsequence $\{\psi_n\}$ such that (for some constructible constants $C_1,C_2>0$) $\max\{\|m_n\alpha\|,\|m_n\beta\|\}<\frac{C_1}{m_n^{1/2}}$ and $\|\psi_n\alpha\|<\frac{C_2}{\psi_n^{1/2}\log \psi_n}$ for all $n$. As a consequence, we have $\psi_n\|\psi_n\alpha\|\|\psi_n\beta\|<\frac{C_1 C_2}{\log \psi_n}$, thus giving an effective proof of Littlewood's conjecture for the pair $(\alpha,\beta)$. Our proofs are elementary and use only standard results from algebraic number theory and the theory of continued fractions.