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Arithmetic classification and non-Pisot singularity for Bernoulli convolutions
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Abstract
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.
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Follow-up works
- Guozheng Cheng, Xiang Fang, Xueqing Ma, Hongli Zhang, Strict correlation-dimension drop for quartic Salem Bernoulli convolutions (arXiv:2610.09902, 7 October 2026). Proves correlation dimension strictly less than one (so no $L^2$ density) for $\nu_{\beta^{-1}}$ at every quartic Salem $\beta\in(1,2)$; it cites this manuscript as settling the measure type for these parameters and says its collision-based argument is different and is not implied by the OpenAI result.
- Guozheng Cheng, Xiang Fang, Xueqing Ma and Hongli Zhang, Strict Hausdorff-dimension drop for Quartic Salem Bernoulli convolutions (arXiv:2610.12072v2, 9 October 2026). Reports Hausdorff dimension strictly below one for every quartic Salem reciprocal and explicit deficit bounds for the two overlapping parameters, citing OpenAI's singularity theorem only for comparison and stating that its proofs and computations do not rely on that manuscript.
Related prior literature mentioned by others
- De-Jun Feng, Multifractal analysis of Bernoulli convolutions associated with Salem numbers (arXiv:1111.2414, 10 November 2011; comments say accepted by Adv. Math.). Cheng, Fang, Ma and Zhang's correlation-dimension paper cites it as prior work on local dimensions for the Salem family of largest roots of $x^n-x^{n-1}-\dots-x+1$, $n\ge4$; for $n=4$ this is $x^4-x^3-x^2-x+1$, one of the paper's two explicit examples (Corollary sal:examples).
- Victor Kleptsyn, Mark Pollicott and Polina Vytnova, Uniform lower bounds on the dimension of Bernoulli convolutions (arXiv:2102.07714, revised 17 January 2022). Cheng, Fang, Ma and Zhang's correlation-dimension paper cites Appendix A.1 for rigorous correlation-dimension lower bounds at both quartic Salem parameters treated in the OpenAI manuscript.
Possibly relevant literature (found by the agent, not checked)
- Nikita Sidorov, Bernoulli convolutions -- 2023 (arXiv:2311.00569, v6 December 2023). An unrefereed preprint whose abstract claims that for Salem $\theta$ the measure $\mu_\theta$ is equivalent to Lebesgue measure on its support, which conflicts with this paper's singularity theorem for quartic Salem parameters.
- Karma Dajani, Charlene Kalle, Singular non-Pisot Bernoulli convolutions (arXiv:1708.05544, 2017). An earlier claim of explicit singular Bernoulli convolutions at Salem and other non-Pisot parameters in $(1,2)$; it was withdrawn in v2 with the comment "Mistake in proof."
Digests and expositions
- Latent Seminar, Bernoulli convolutions: measure type and non-Pisot singularity (6 October 2026). Explains the arithmetic approximation criterion and both non-Pisot constructions, while stating that it does not independently verify the complete proofs or execute the degree-31 certificates. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 153
Independent sites; their content is not reviewed here.
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}
}From OpenAI's README for this paper:
Verification
The degree-31 certificate uses Python 3 and only its standard library. From this
paper directory, run without -O or PYTHONOPTIMIZE, since the verifier requires
assertions:
python3 verification/verify_degree31.py
Successful execution exits with status zero and prints
PASS: all degree-31 finite arithmetic checks passed. The script checks the
finite exact-arithmetic certificate; the analytic bounds connecting it to the
infinite products are supplied in the manuscript.
Family 153: Arithmetic classification and non-Pisot singularity for Bernoulli convolutions
Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every $\lambda\in(0,1)$ 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 $(1,2)$, giving examples beyond reciprocal Pisot parameters.
This is the only paper in this family.