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For the coefficients $2^n$ the equidistribution theorem fails. In fact it is easy to exhibit an irrational $a$ such that the sequence $(2^na)_{n=1}^\infty$ is not even dense in $(0,1)$ modulo $1$. For example, take an $a\in (0,1)$ whose binary expansion consists of increasing blocks of $1$'s seperated by $0$'s: $$ a=0.101101110111101111101111110\dots $$