Result 153, Dynamical systems and ergodic theory

Arithmetic classification and non-Pisot singularity for Bernoulli convolutions

Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every λ∈(0,1)\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)(1,2), giving examples beyond reciprocal Pisot parameters.

Classification or exact value

The bigger picture

Why it matters

Repeated fair coin flips can build a random real number by adding positive or negative contributions that shrink geometrically. This manuscript claims an arithmetic way to distinguish when the resulting distribution has a density and when it is singular.

What changes?

The manuscript reports a classification of unbiased Bernoulli convolutions, the distributions of these random sums, for every shrinking factor lambda between zero and one. Absolute continuity means having a density; singularity means being concentrated on a set of zero length. The criterion uses an infinite, one-sided approximation condition involving explicitly defined finite sets of algebraic units, algebraic integers whose reciprocals are also algebraic integers. Despite using finite sets, it is not a finite membership algorithm.

What does that help mathematicians do?

The reported examples show that singularity for parameters between one-half and one is not confined to reciprocals of Pisot numbers. They include reciprocals of every quartic Salem number between one and two, and the reciprocal of a specified non-Pisot root of an explicit degree-31 polynomial. Researchers would therefore have to accommodate these additional arithmetic sources of singularity, rather than treating the familiar Pisot examples as an exhaustive explanation.

Are there practical applications?

The immediate value is foundational: the claimed criterion connects the shape of a probability distribution built from independent coin flips to arithmetic approximation by algebraic numbers. It offers a framework for studying density versus concentration in this family, not a demonstrated computational procedure or a practical method for deciding arbitrary parameter values.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Arithmetic classification and non-Pisot singularity for Bernoulli convolutions

October 3, 2026 36 pages

We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter λ∈(0,1)\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)(1,2) and at the reciprocal of a specified non-Pisot root of an explicit degree-31 polynomial.

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Cite (BibTeX) 
@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}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.