Arithmetic classification and non-Pisot singularity for Bernoulli convolutions

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 2Source 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 PisotSource 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 fadeSource 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 1Source 4. The law is singular exactly when such units approach λ from one side at a fixed geometric rateSource 5. This is an infinite approximation conditionSource 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.