In this paper, we discuss a problem of multiplicative Diophantine approximation of the expansions under different bases restricted on a line. More specifically, let φ : ℕ → [0,∞) be a positive function. A dichotomy law of the Hausdorff measure for the following set: M(φ) = {x ∈ [0, 1] : ∥2nx∥⋅∥3nx∥ < φ(n) for i.m. n ∈ ℕ}, is obtained, which depends the convergence or divergence of certain series.

中文翻译：

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一条线上不同基数下展开的倍增二苯胺逼近

在本文中，我们讨论了在一条线上限制的不同基下展开的乘法丢番图逼近问题。更具体地说，让φ ： ℕ → [0,∞)成为正函数。下列集合的豪斯多夫测度二分法： 米(φ) = {X ∈ [0, 1] ： ∥2nX∥⋅∥3nX∥ < φ(n)对于我n ∈ ℕ}, 得到，这取决于某些系列的收敛或发散。