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Singular Bernoulli convolutions at non-Pisot parameters

For 0<λ<10<\lambda<1 the Bernoulli convolution νλ\nu_\lambda is the law of ∑n≥0±λn\sum_{n\ge0}\pm\lambda^n with independent fair signs; it is either singular or absolutely continuous (Jessen-Wintner). It is singular for λ<1/2\lambda<1/2, and Erdos (1939) proved it singular whenever 1/λ1/\lambda is a Pisot number. Salem (1943) showed the Fourier transform tends to zero for every other λ\lambda, so Fourier nondecay cannot detect singularity there. Solomyak proved absolute continuity for almost every λ∈(1/2,1)\lambda\in(1/2,1) and Shmerkin showed the exceptional set has Hausdorff dimension zero, but no singular example was known with 1/λ1/\lambda not Pisot, for instance a Salem number. Is there λ∈(1/2,1)\lambda\in(1/2,1) with 1/λ1/\lambda not a Pisot number for which νλ\nu_\lambda is singular?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Fractal geometry and harmonic analysis: Bernoulli convolutions
Posed by
Recorded as open by Baker, Koivusalo, Troscheit and Zhang (J. London Math. Soc. 2026) and Kern and Sterk (arXiv:2608.14155, 2026), as cited; background Erdos (1939) and Salem (1943)
Year posed
—
Years open
—
Solved
2026-10-03
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Two non-Pisot singularity theorems. If β∈(1,2)\beta\in(1,2) is a Salem number of degree four then ν1/β\nu_{1/\beta} is singular; explicit cases are the real roots of x4−x3−x2−x+1x^4-x^3-x^2-x+1 and x4−2x3+x2−2x+1x^4-2x^3+x^2-2x+1. An explicit degree-31 polynomial has a unique real root β∈(1.8392,1.8393)\beta\in(1.8392,1.8393), not Pisot, with ν1/β\nu_{1/\beta} singular. The manuscript also proves a criterion: νλ\nu_\lambda is singular iff λ\lambda lies in an explicit limsup set of one-sided approximations by finite sets of algebraic units; the paper says this is an infinite approximation condition, not a finite membership test, so it is not an effective classification. It does not decide Salem numbers of degree six or more, nor which non-Pisot parameters are singular in general.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and the statements of the arithmetic criterion, the quartic Salem theorem and the degree-31 theorem in the TeX source, read against the existence question as the manuscript cites it; the proofs were not refereed. The degree-31 example rests on a finite exact-arithmetic certificate (verification/verify_degree31.py, standard-library Python) plus analytic bounds in the manuscript; the certificate was not re-run here. The quartic Salem result is a pure argument. The release has no Lean formalization for this family, and the README cautions that unformalized results could have issues.

Sources

Changelog1 change

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