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.
Hausdorff dimension of sets with restricted, slowly growing partial quotients
I. J.Good（1941）证明了$(0,1)$中的无理数集合如果其偏导数$a_n$趋于无穷大，则它具有豪斯多夫维数$1/2$。一些相关的结果施加限制的类型$a_n\in B$或$a_n\geq f(n)$，其中$B$是一个无限子集的$\mathbb N$和$f$是一个快速增长的功能与$n$。我们证明了，对于任意的$B$和任意的$f$，其值在$[\min B,\infty)$中并趋于无穷大，$(0,1)$中的无理数集合使得\[ a_n\in B,\ a_n\leq f(n)\text{ for all $n\in\mathbb N$, and }a_n\to\infty\text{ as }n\to\infty\]具有豪斯多夫维数$τ(B)/2,$，其中$τ(B)$是$B$的收敛指数.
University of Arkansas（阿肯色大学）；School of Mathematics, Fry Building, Woodland Road, Bristol BS8 1UG, United Kingdom（布里斯托大学数学学院）；School of Mathematical Sciences, Mathematical Sciences Building, University Park, Nottingham, NG7 2RD, United Kingdom（诺丁汉大学数学科学学院）
Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers
The Graduate Center at the City University of New York（纽约市立大学研究生中心）；Oxford University（牛津大学）；University College, Oxford（牛津大学学院）；College of Staten Island（史泰登岛学院）
