Constructing Simultaneous Diophantine Approximations of Certain Cubic Numbers

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.