Irrational self-similar sets

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 $K$ zero, then for almost every $t$, $$K+t=\{x+t:x\in K\}$$ only consists irrational or transcendental numbers. In this note, we shall consider some classes sets, and explicitly construct such $t$'s. Our main idea from $q$-expansions.