TITLE: A point set of power series with coefficients in {-1, 1}. Connected or not?
QUESTION [27 upvotes]: Let $z$ be a fixed complex number with $|z|<1$ and consider the set
$$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$
What can be said about the set $M$ of those $|z|\lt 1$ such that $X_z \subset \mathbb{C}$ connected?

REPLY [7 votes]: What you are referring to is called the connectedness locus. The most recent important paper on the subject is probably this paper by Calegari, Koch and Walker. 
In particular, they show that the connectedness locus contains infinitely many holes. The paper also contains an extensive list of references.