# Arithmetic classification and non-Pisot singularity for Bernoulli convolutions

> Claimed singular Bernoulli convolutions beyond Pisot reciprocals, plus an arithmetic test for every parameter. Plain-English explainer with sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-153/
Last updated: 2026-10-11

**Claimed singular Bernoulli convolutions beyond Pisot reciprocals, plus an arithmetic test for every parameter.**

**Trust:** Claimed, not checked · **Papers:** 1 · **First paper:** October 3, 2026 · **Field:** Dynamical systems and ergodic theory · **Entry:** 153

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

Add up ±1 ± λ ± λ² and so on, choosing each sign by a fair coin flip. The result has a density or sits on a set of zero length, and for many λ it was unknown which. Above one half, every known zero-length case had 1/λ equal to a Pisot number, following Erdős in 1939. This paper claims new zero-length cases, at one over each degree-four Salem number between 1 and 2, and at one more number. It also claims an exact arithmetic test for every λ, though not a finite procedure.

## At a glance

- **First step:** 1939 (Erdős, singularity when 1/λ is a Pisot number)
- **Significance:** Major. Singular examples beyond Pisot reciprocals are recorded as open in the recent literature the paper cites.
- **Trust:** Claimed, not checked
- **Field:** Dynamical systems and ergodic theory (1 paper)

## The explanation

Flip a fair coin again and again. Heads adds a plus sign, tails a minus sign. Now add ±1 ± λ ± λ² ± λ³ and so on, where λ is a fixed number between 0 and 1. The sum is a random number. Its probability law is called the unbiased Bernoulli convolution for λ. Take λ = 1/2. Then the sum is spread perfectly evenly over the interval from −2 to 2.

Now take λ = 1/3. Fix the first ten signs, which allows 1,024 choices. The rest of the sum then stays inside a window of width 3 × (1/3)¹⁰. So all outcomes sit inside 1,024 short intervals. Their total length is 3 × (2/3)¹⁰, about 0.052. With more flips that total keeps shrinking towards zero. The law is singular: all its probability sits on a set of zero total length. This simple covering argument works for every λ below 1/2.

The hard range is λ between 1/2 and 1. Since 1935 it has been known that each such law has a pure type. It is either singular or absolutely continuous, meaning it has a density, like a smooth histogram. The classification problem asks which type holds for each λ.

Number theory enters here. A Pisot number is a number above 1 that solves a polynomial with whole-number coefficients and leading coefficient 1. Its other roots must all have size below 1. The golden ratio, about 1.618, is one. Its partner root is about −0.618. In 1939 Erdős proved the law is singular when 1/λ is a Pisot number. So λ ≈ 0.618 gives a singular law. In 1962 Garsia found a different class with a density. In 1995 Solomyak showed that almost every λ in the range has a density. In 2014 Shmerkin showed the possible exceptions form a very thin set, of dimension zero. Still, the recent literature lists one gap as open. No singular example was known whose 1/λ is not a Pisot number.

Salem numbers are the natural next suspects. A Salem number is like a Pisot number, but its other roots are 1/β and roots of size exactly 1. Take x⁴ − x³ − x² − x + 1. Its largest root is about 1.7221. Its other roots are about 0.5807, and a pair of complex roots of size exactly 1. In 1943 Salem showed that a key tool fails here. That tool, the Fourier transform, measures how strongly the law repeats at each frequency. For non-Pisot parameters it fades to zero, so it cannot prove singularity.

The paper claims to prove singularity at 1/β for every degree-four Salem number β between 1 and 2[1]. That includes λ ≈ 1/1.7221 ≈ 0.5807 from the example. It claims the same for one root of an explicit degree-31 polynomial, between 1.8392 and 1.8393, which is not Pisot[2]. For the Salem case, it builds certain wave-like averages from the Fourier transform. Under the Bernoulli convolution they stay concentrated, while under ordinary length they fade[3]. That difference shows the law is singular.

The paper also claims a criterion for every λ. It uses finite lists of algebraic units, roots of polynomials with coefficients −1, 0 or 1[4]. The law is singular exactly when such units approach λ from one side at a fixed geometric rate[5]. This is an infinite approximation condition[6]. It need not give a short test for a given number.

How sure is this? No Lean formalization is listed for this result in the openai/math release. The degree-31 case comes with a Python script that checks its finite exact arithmetic. The paper supplies the analytic bounds linking that check to the infinite products in the proof. No specialist review has been published.

**What it doesn't show.** This does not give a finite procedure that decides, for a given λ, which of the two cases holds.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Major only if the proofs hold. No Lean formalization is listed and no specialist review is published. A script checks only the finite arithmetic of the degree-31 case.

## Words used

- **Bernoulli convolution:** The probability law of ±1 ± λ ± λ² ± …, with each sign chosen by a fair coin.
- **Singular:** Putting all probability on a set of zero total length, so no density exists.
- **Absolutely continuous:** Having a density, so probability is spread like a histogram with no zero-length pile-ups.
- **Pisot number:** A root above 1 of a whole-coefficient polynomial with leading coefficient 1, whose other roots all have size below 1.
- **Salem number:** Like a Pisot number, but its other roots are 1/β and roots of size exactly 1.
- **Fourier transform:** A measure of how strongly a probability law repeats at each frequency.
- **Algebraic unit:** A root of a whole-coefficient polynomial whose leading and constant coefficients are each 1 or −1.

## The original paper

- [Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf) - October 3, 2026

## Quoted lines from the paper (6)

[1] From [Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf):

> Let $\beta\in(1,2)$ be a Salem number of degree four.

[2] From [Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf):

> The polynomial \(p\) in (…) has a unique real root \(\beta\in(1.8392,1.8393)\). This root is not a Pisot number, and \(\nu_{1/\beta}\) is singular with respect to Lebesgue measure.

[3] From [Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf):

> Their averages concentrate under the Bernoulli convolution while tending to zero in Lebesgue mean square.

[4] From [Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf):

> Their minimal polynomials have coefficients in $\{-1,0,1\}$, prescribed degree, and irreducible reduction modulo two.

[5] From [Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf):

> Singularity is equivalent to one-sided approximation by these retained targets at a fixed geometric rate.

[6] From [Arithmetic classification and non-Pisot singularity for Bernoulli convolutions](https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf):

> 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.

## Other sources

- [OpenAI, “Arithmetic classification and non-Pisot singularity for Bernoulli convolutions”, openai/math release preprint (October 3, 2026)](https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf): “We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter $\lambda\in(0,1)$.”
- [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”
- [B. Jessen and A. Wintner, “Distribution functions and the Riemann zeta function”, Transactions of the AMS 38 (1935), 48–88](https://doi.org/10.1090/S0002-9947-1935-1501802-5)
- [P. Erdős, “On a family of symmetric Bernoulli convolutions”, American Journal of Mathematics 61 (1939), 974–976](https://doi.org/10.2307/2371641)
- [R. Salem, “Sets of uniqueness and sets of multiplicity”, Transactions of the AMS 54 (1943), 218–228](https://doi.org/10.1090/S0002-9947-1943-0008428-8)
- [A. M. Garsia, “Arithmetic properties of Bernoulli convolutions”, Transactions of the AMS 102 (1962), 409–432](https://doi.org/10.1090/S0002-9947-1962-0137961-5)
- [B. Solomyak, “On the random series ∑±λⁿ (an Erdős problem)”, Annals of Mathematics 142 (1995), 611–625](https://doi.org/10.2307/2118556)
- [P. Shmerkin, “On the exceptional set for absolute continuity of Bernoulli convolutions”, Geometric and Functional Analysis 24 (2014), 946–958](https://doi.org/10.1007/s00039-014-0285-4)
- [openai/math, verification README for the degree-31 certificate (2026)](https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/README.md): “The script checks the finite exact-arithmetic certificate; the analytic bounds connecting it to the infinite products are supplied in the manuscript.”

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