Littlewood's conjecture that given any real numbers \(\alpha, \beta\), \(\liminf_{q\to \infty}q\|q\alpha \|\|q\beta \|=0\) \((\|\alpha\|= \min\{|x-k|: k\in\mathbb{Z}\})\) has resisted resolution for many years. The conjecture holds if either \(\alpha\) or \(\beta\) are not badly approximable (\(\alpha\) is badly approximable if \(q\|q\alpha\|\geq c\) for some constant \(c\) and all \(q\in\mathbb{N}\), this is equivalent to the partial quotients in the continued fraction expansion for \(\alpha\) being bounded). It is shown that for each badly approximable \(\alpha\), the set \({\mathbf G} (\alpha)\) of badly approximable numbers satisfying \(\|q_n\beta \|0\). This is nevertheless the first significant advance since 1955 when Cassels and Swinnerton-Dyer proved the conjecture held for any pair of cubic irrationals in the same cubic field. The result is obtained by establishing the existence of a suitable measure \(\mu\) on the Cantor-type set \(F_N\) of numbers with partial quotients at most \(N\) and invoking the mass distribution principle. The measure needed had been constructed by \textit{R. Kaufman} [Mathematika 27, 262-267 (1980; Zbl 0455.10035)]. The hard part of the proof, which uses the pair-wise independence generalisation of the Borel-Cantelli lemma, is to show that \(\mu(G_N (\alpha)) >0\), where \(G_N(\alpha)= G(\alpha) \cap F_N\). The paper concludes with a nice application of the Davenport-Erdős-LeVeque theorem to prove that for \(\mu\)-almost all \(\beta\), the set \(\{q_n\beta: n\in\mathbb{Z}\}\) is uniformly distributed; whence \(\{\beta\in F_N:\lim \inf_{n\to \infty}\|q_n\beta \|= 0\}\) has \(\mu\)-measure 1 and so Hausdorff dimension 1.