Can anyone provide a proof of the equidistribution theorem using Weyl's criterion for the case of $c\*a \,\,\, \text{(mod 1)}$ where $c=2^n: \,\,\, \forall n \in N\_0$ for irrational algebraic $a$? The original equidistribution theorem is for the case that $c = n: \,\,\, \forall n \in N\_0$. Much thanks in advance.



Proof of equidistribution theorem for exponential coefficients
You are asking whether a number $a$ is a normal number in base 2 or not. This is an extremely complicated problem, and very far from being solved. Generally speaking, there exist several constructions of normal numbers, but we don't have any tool to determine whether a given number is normal or not. It is (sometimes) conjectured that all algebraic irrationals are normal, but a proof of such a statement is very, very, very, very, very far away.


You may be interested in the paper 


Bailey, D.; Crandall, R.E.: On the random character of fundamental constant expansions. Experiment. Math. 10 (2001), no. 2, 175–190.


where this topic is discussed. 


(By the way, the topic has a connection with ergodic theory. You are looking at the orbit of $a$ under the map $T:~x \mapsto 2x \mod 1$.)


