Fractal sets only contain irrationals.

In this paper, we prove that many fractals sets generated by the associated dynamical systems only contain irrationals. We give two applications. Firstly, we explicitly construct some overlapping self-similar sets which only consists of irrationals. Secondly, we show that the translations of some univoque sets are subsets of irrational numbers. Our proof mainly relies on the fractional parts of powers of some real numbers.