I am interested in understanding the properties of the following set $S$.
Given a convergent series $\sum_{n\ge0}a_n$ of complex numbers, let us define $S$ by $$S=\left\{\sum_{n=0}^\infty\epsilon_n a_n \ : \,\epsilon\in\{0,1\}^\mathbb{N}\right\}.$$
I do know that if $\forall n\in\mathbb{N}$, $a_n\ge0$, and if $$\forall n\in\mathbb{N}, \ \,a_n\le\sum_{k=n+1}^\infty a_k \ ,\tag{$\star$}$$ then $S=[0,A]$, where $A=\sum_{n=0}^\infty a_n$ by definition.
The condition $(\star)$ is satisfied if for example $\forall n\in\mathbb{N},\,a_n=q^n$ with $\frac12\le q<1$.
But what happens if $\forall n\in\mathbb{N},\,a_n=q^n$ with $0<q<\frac12$ ?
And more generally, what can be said about the set $S$? I suspect some fractal structure, but I am not sure. I only know how to prove that $S$ is a compact subset of $\mathbb{C}$.
Any hints and/or references would be appreciated !