Personal question : Let $z_0$ be a $2^k$th root of unity. We obtain for the function $f(z)=\sum_{n \geq 0} z^{2^n}$ (radius of convergence $R=1$) that $f(z_0)=\sum_{n=0}^{k-1}z_0^n + 1+1+ \dots$. Why all such points is dense on the unit circle?
I tried to obtain results from that question, but I blocked. Is anyone could help me at this point?
The $2^k$-th roots of unity are $e^{2\pi i \lambda},$ where $\lambda = \frac{a}{2^b}$ is a dyadic rational number. It's enough to know that dyadic rational numbers are dense in $[0,1]$. In other words, that any number in $[0,1]$ can be written as an infinite "decimal" expansion in base $2$.