In this paper we obtain new lower bounds for the upper box dimension of αβ sets. As a corollary of our main result, we show that if α is not a Liouville number and β is a Liouville number, then the upper box dimension of any αβ set is 1. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.

中文翻译：

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αβ集合的新维度界限

在本文中，我们为αβ集的上框维度获得了新的下界。作为我们主要结果的推论，我们证明如果α不是 Liouville 数并且β是 Liouville 数，那么任何αβ集合的上框维度都是 1。我们还使用我们的维度边界来获得仿射嵌入的新结果自相似集。