On the Equidistribution Weyl Theorem for $\{2^kx\}$

Question

It is well known that the sequence $\{2^k \eta\} \bmod 1$ is uniformly distributed for almost all, but not all, irrational $\eta$ in $(0,1)$. If I fix an irrational number $0<x<1$ ($x$ is actually of the following form $x=0.0100100101000000100001\dots$, i.e. many zeros and some ones), is it possible to say something about the distribution of $\{2^k x\}$ in $(0,1)$?

Answer 1

As you say, it is uniformly distributed for almost every $x$ but not for every $x$. What you can say about a particular $x$ will depend on what you know about $x$. In particular, what is important is the base-$2$ expansion of $x$.