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.

Manuscript

Arithmetic classification and non-Pisot singularity for Bernoulli convolutions

October 3, 2026 · source · cite as \cite{OAI:Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026}

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.

Result 153 · Dynamical systems and ergodic theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.