Set characterization

Question

I need to understand what are the elements of the set $\left[0,1\right]$ whose non-terminating decimal expansions contain only the digits 3, 5 and 7.

I suppose it is an auto-similar set, up to an extent, but I cannot really figure out how I can build it with analytic processes.

Thanks for your help

Answer 1

It appears that $A=(\frac1{10}A+0.3)\cup(\frac1{10}A+0.5)\cup(\frac1{10}A+0.7)$.

Answer 2

Hagen von Eitzen’s answer gives you the key idea. To implement it, let $A_0=[0,1]$, and for $n\in\Bbb N$ let

$$A_{n+1}=\left(\frac1{10}A_n+0.3\right)\cup\left(\frac1{10}A_n+0.5\right)\cup\left(\frac1{10}A_n+0.7\right)\;.$$

Then set $A=\bigcap_{n\in\Bbb N}A_n$.