Singular Bernoulli convolutions

Singular Bernoulli convolutions are the subject of a classical problem: for which λ in (1/2, 1) is the distribution of the random series Σ ±λⁿ, with independent fair signs, singular rather than absolutely continuous? Erdős showed singularity when 1/λ is a Pisot number. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 3 October 2026, claims an arithmetic criterion that classifies every parameter, and singularity at the reciprocals of every quartic Salem number in (1, 2) and at one further non-Pisot parameter. The family has no Lean formalization. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

As recounted in the introduction of the manuscript:
- Bernoulli convolutions are of pure type (Jessen and Wintner). For λ < 1/2 they are singular and at λ = 1/2 uniform; Erdős proved singularity when λ⁻¹ is Pisot, by showing that the Fourier transform does not tend to zero, and Salem showed that it does tend to zero otherwise, so Fourier nondecay cannot detect other singular parameters.
- Solomyak proved absolute continuity, with an L² density, for almost every λ in (1/2, 1); Hochman related dimension loss to strong concentration of cylinder positions, and Shmerkin showed that the possible singular parameters form a set of Hausdorff dimension zero. Garsia connected singularity with the entropy of finite sums, and Breuillard and Varjú showed that dimension-deficient parameters are exceptionally well approximated by algebraic parameters of the same kind.

The original results

Claim. The classification theorem of Arithmetic classification and non-Pisot singularity for Bernoulli convolutions states that for every λ in (0, 1) the Bernoulli convolution is singular exactly when λ satisfies an infinite one-sided approximation condition by explicit finite sets of real algebraic units: units whose minimal polynomials have coefficients in {−1, 0, 1}, prescribed degree and irreducible reduction modulo two. The manuscript describes these as examples beyond reciprocal Pisot parameters. It also proves singularity for the reciprocal of every quartic Salem number in (1, 2), and for a root near 1.8393 of an explicit polynomial of degree 31 with a pair of complex conjugate roots outside the unit circle.
Method, as described: for the criterion, concentration of the measure produces many pairs of finite bit words with nearby sums, whose differences are completed to height-one polynomials with a root above the parameter; a uniform nonvanishing estimate and prime polynomials in arithmetic progressions over F₂[t] keep the required sign change. For quartic Salem parameters, algebraic multipliers whose unit-circle conjugates decay faster than the cosine losses accumulate, together with trace identities, localize many Fourier phases. The degree-31 case uses a lattice whose box volume grows slightly faster than the reciprocal parameter to the word length.
Source. The manuscript Arithmetic classification and non-Pisot singularity for Bernoulli convolutions at commit 1581d4c.
Formalization. The repository has no Lean scope note for this family.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 20 October 2026)
2026-10-07	G. Cheng, X. Fang, X. Ma and H. Zhang, Strict correlation-dimension drop for quartic Salem Bernoulli convolutions, arXiv:2610.09902v1	Stronger result (different method)	For every quartic Salem number β in (1, 2), the correlation dimension of the Bernoulli convolution at β⁻¹ is strictly less than one, so neither measure has an L² density; computer-assisted bounds give 0.99999 < D₂ < 1 − 10⁻⁶⁵ and 0.99999 < D₂ < 1 − 10⁻⁷⁵ for the two parameters. The paper states that during its final preparation OpenAI released a manuscript proving singularity for these parameters, which settles the measure-type question but does not imply the correlation-dimension drop, and that the arguments differ.	arXiv preprint, version 1 of 7 October 2026. The paper declares use of an OpenAI model for language editing, formatting and limited assistance with local reasoning and numerical verification.
2026-10-08	G. Cheng, X. Fang, X. Ma and H. Zhang, Strict Hausdorff-dimension drop for Quartic Salem Bernoulli convolutions, arXiv:2610.12072	Stronger result	For every quartic Salem number β, the Bernoulli convolution at β⁻¹ has Hausdorff dimension strictly less than one, which implies singularity; for the two parameters in (1, 2), exact calculations give deficits of about 1.4·10⁻²³ and 7.8·10⁻²². The proof turns large Fourier coefficients at separated frequencies into an entropy deficit and is independent of the numerical part. The paper cites the OpenAI manuscript in its references.	arXiv preprint, version 2 of 9 October 2026; verification programs in exact rational interval arithmetic accompany it. The paper makes the same AI-use declaration.
2026-10-16	J. Jardine, Lattice24, referee review in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Reports no gap and states that the reviewer read all sections and the certificate appendix. Its one minor note is that in the degree-31 argument a coupling between finite and stationary truncated characters should start a few dozen indices later than stated, which affects only a bounded number of boundary indices and not the conclusion.	Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.

Notes

The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.