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.
Cite
@misc{OAI:Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026,
author = {{OpenAI}},
title = {{Arithmetic classification and non-Pisot singularity for Bernoulli convolutions}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf}{OAI:Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026}},
year = {2026}
}