论文标题
非理性的自相似集
Irrational self-similar sets
论文作者
论文摘要
令$ k \ subset \ mathbb {r} $为$ \ mathbb {r} $上定义的自相似集。很容易证明，如果$ k $的lebesgue度量为零，那么对于勒布斯格来说，几乎每$ t $，$$ k+t = \ {x+t = \ {x+t：x \ in K \} $$仅由非理性或先验数字组成。在本说明中，我们将考虑一些类似的自相似集，并明确构建这种$ t $。我们的主要想法来自$ q $ - 扩展。
Let $K\subset\mathbb{R}$ be a self-similar set defined on $\mathbb{R}$. It is easy to prove that if the Lebesgue measure of $K$ is zero, then for Lebesgue almost every $t$, $$K+t=\{x+t:x\in K\}$$ only consists of irrational or transcendental numbers. In this note, we shall consider some classes of self-similar sets, and explicitly construct such $t$'s. Our main idea is from the $q$-expansions.
