We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full‐dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong form of a result of Haynes et al. Finally, we establish a similar result involving inhomogeneously badly approximable numbers, making progress towards a problem posed by Pollington et al.

中文翻译：

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完全近似数的完全不均匀的乘以二磷黄嘌呤近似

我们建立了具有不均匀移位的Littlewood猜想的强形式，即在垂直线上有一对全维的极差近似数对。我们还证明了这种性质的统一主张，归纳了Haynes等人的结果的强形式。最后，我们建立了一个相似的结果，涉及非均质极差的近似数字，从而朝着Pollington等人提出的问题取得了进展。