Arithmetic classification and non-Pisot singularity for Bernoulli convolutions

What OpenAI claims

Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every λ ∈ (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.

Paper

1. Arithmetic classification and non-Pisot singularity for Bernoulli convolutions

   3 October 2026 · Preprint, not peer reviewed · PDF

   We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter λ ∈ (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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.