Arithmetic classification and non-Pisot singularity for Bernoulli convolutions
We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter $\lambda\in(0,1)$. The criterion is expressed through one-sided approximation by explicitly defined finite sets of algebraic units; it is an infinite approximation condition, not a finite membership algorithm. We also establish singular examples beyond reciprocals of Pisot numbers: singularity holds at the reciprocal of every quartic Salem number in $(1,2)$ and at the reciprocal of a specified non-Pisot root of an explicit degree-31 polynomial.
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