Consider the class of power series with coefficients in {-1,0,1}. It is well known that the distribution of zeros of the functions of this class is related to several problems in fractal geometry. I will review those connections and then present some new results in the area.
Our main result is the following: let $\Omega_2$ be the set of all real numbers in (0,1) which are roots of multiplicity at least 2 of some power series with coefficients in {-1,0,1}. The set $\Omega_2$ is disconnected; in fact, it has at least 58 connected components (we conjecture that in fact it has infinitely many connected components).
The proofs are mostly computer-assisted, so the talk will include a discussion of several algorithms and numerous pictures.
This is joint work with Boris Solomyak.