I want to take this opportunity to give an application of algorithmic randomness theory to this area. Accidentally I am working on the similarity problem recently and found some interesting applications of algorithmic randomness theory to the area.
For the notation of randomness you may refer to the book by Downey-Hirschfeldt or Nies.
Proof: It is simple to see that if $\delta$ is $x$-random, then so is $\delta2^{-n}$ for any $n\geq 1$. But $E$ is conull and so there must be a real $x$ so that every $x$-random belongs to $E$. Then it is clear that every $x$-random real $\delta$ meets your requirement. QED
Actually this method can be used to prove that for any countable set $A\subset (0,1)$ , there is a real $\delta$ so that $\delta y\in E$ for any $y\in A$.