Irrational self-similar sets

Let $K\subset\mathbb{R}$ be a self-similar set defined on $\mathbb{R}$. It is well-known 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 the numbers which are irrational or even transcendental. In this note, we shall consider some classes of self-similar sets, and explicitly construct such $t$. Our main idea is from the $q$-expansions.