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Zeros of \(\{-1,0,1\}\) power series and connectedness loci for self-affine sets. (English) Zbl 1122.30002

Summary: We consider the set \(\Omega_2\) of double zeros in \((0,1)\) for power series with coefficients in \(\{-1,0,1\}\). We prove that \(\Omega_2\) is disconnected, and estimate \(\min \Omega_2\) with high accuracy. We also show that \([2^{-1/2}-\eta,1)\subset \Omega_2\) 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.

MSC:

30B10 Power series (including lacunary series) in one complex variable
28A80 Fractals
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30-04 Software, source code, etc. for problems pertaining to functions of a complex variable
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