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 $C_1,C_2>0$) $\max \{\mathrm{\parallel }{m}_n\alpha \mathrm{\parallel },\mathrm{\parallel }{m}_n\beta \mathrm{\parallel }\}<\frac{C_1}{m_n^{1\mathrm{/}2}}$ and $\mathrm{\parallel }{\psi }_n\alpha \mathrm{\parallel }<\frac{C_2}{{\psi }_n^{1\mathrm{/}2}\log {\psi }_n}$ 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.