Exceptional singular Bernoulli convolutions beyond Pisot parameters

Determine whether there exist parameters \(\beta\in(1/2,1)\) other than reciprocals of Pisot numbers for which the corresponding Bernoulli convolution is singular, thereby characterizing the exceptional parameters governing absolute continuity and singularity.

Background

The paper relates the stationary distribution of the Bernoulli-driven $AR(1)$ process to a Bernoulli convolution. In the symmetric case, Bernoulli convolutions are known to be singular for parameters below $1/2$, for reciprocals of Pisot numbers in $(1/2,1)$, and absolutely continuous for Lebesgue-almost-every parameter in $(1/2,1)$. The unresolved issue is whether singular examples exist in $(1/2,1)$ beyond the known reciprocal-Pisot cases. The authors explicitly state that their results do not address this characterization problem.