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

Abstract

We consider the set Ω₂ of double zeros in (0, 1) for power series with coefficients in {-1, 0, 1}. We prove that Ω₂ is disconnected, and estimate minΩ₂ with high accuracy. We also show that [2^-1/2 - n, 1) ⊂ Ω₂ for some small, but explicit, n > 0 (this was known only for n = 0). These results have applications in the study of infinite Bernoulli convolutions and connectedness properties of self-affine fractals.

Keywords:

• Primary 30C15
• Secondary 28A80
• Zeros of power series
• self-affine fractals