Zeros of $\{-1,0,1\}$ Power Series and Connectedness Loci for Self-Affine Sets

Abstract

We consider the set $\Om_2$ of double zeros in $(0,1)$ for power series with coefficients in $\{-1,0,1\}$. We prove that $\Om_2$ is disconnected, and estimate $\min \Om_2$ with high accuracy. We also show that $[2^{-1/2}-\eta,1)\subset \Om_2$ for some small, but explicit, $\eta>0$ (this was known only for $\eta=0$). These results have applications in the study of infinite Bernoulli convolutions and connectedness properties of self-affine fractals.