Arithmetic classification and non-Pisot singularity for Bernoulli convolutions. Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every $\lambda\in(0,1)$ by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in $(1,2)$, giving examples beyond reciprocal Pisot parameters.
released 2026-10-03 | 3 theorems · 12 lemmas · 21 proofs · 13,212 words |
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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.