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Math-2440 · mathematics · Analysis

Absolute Continuity and Densities of Bernoulli Convolutions

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For $1/2<\lambda<1$, let $\nu_\lambda$ be the distribution of $\sum_{n\geq0}\varepsilon_n\lambda^n$, where the $\varepsilon_n$ are independent fair signs. Determine exactly when $\nu_\lambda$ is absolutely continuous, and classify the possible regularity of its density.

Formulation references: [S1], [S2].

Short English statement

For $1/2<\lambda<1$, let $\nu_\lambda$ be the distribution of $\sum_{n\geq0}\varepsilon_n\lambda^n$, where the $\varepsilon_n$ are independent fair signs. Determine exactly when $\nu_\lambda$ is absolutely continuous, and classify the possible regularity of its density.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2440-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2440

Manuscript 1 (family 153, 2026-10-03): The arithmetic criterion classifies absolute-continuity parameters throughout (1/2,1), but does not classify all possible density regularities, the second clause of this bundle.

Absolute Continuity and Densities of Bernoulli Convolutions

OpenAI's claim

Unbiased Bernoulli convolutions are claimed to admit an arithmetic singularity criterion for every parameter, with singular reciprocal quartic-Salem and specified degree-31 non-Pisot examples. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $\nu_\lambda$ be the law of $\sum_{j\ge0}\epsilon_j\lambda^j$ for independent unbiased signs $\epsilon_j\in\{-1,1\}$ and $0<\lambda<1$. OpenAI claims an exact arithmetic singularity criterion. Define $R(n,d)$ as the real algebraic units $r\in(1/2,1)$ of degree $dn$ whose minimal polynomial has coefficients in $\{-1,0,1\}$ and is irreducible modulo two; choose its sign to make the constant term $-1$. Let $\pi_n(p)$ be its first $n$ coefficient vector and \[ W_n(r;d)=\{\pi_n(p_q):q\in R(n,d),\ |q-r|\le4^{-n}\},\quad \mathcal O_u=\{w:(2u_i-1)w_i\ge0\}, \] with $u\in\{0,1\}^n$. Let $R_*(n,d,j)$ contain those $r$ for which at least $2^n/d$ orthants contain at least $j(2r)^n$ vectors of $W_n(r;d)$. It claims \[ \nu_\lambda\text{ singular}\quad\Longleftrightarrow\quad \lambda\in(0,1/2)\cup\left((1/2,1)\cap\bigcup_{d\ge2}\bigcap_{j\ge1}\limsup_{n\to\infty}\bigcup_{r\in R_*(n,d,j)}[r-4^{-n},r)\right), \] and absolute continuity exactly on the complement; replacing limsup with liminf leaves the set unchanged. It separately claims singularity at $\lambda=1/\beta$ for every degree-four Salem number $\beta\in(1,2)$, and for the unique real root $\beta\in(1.8392,1.8393)$ of \[ t^{31}-2t^{30}+\sum_{j=0}^9(t^{3j+1}-t^{3j}), \] which it claims is non-Pisot. Pisot means an algebraic integer greater than one with all other conjugates strictly inside the unit circle; a degree-four Salem number has the other conjugates $\beta^{-1}$ and a conjugate pair on that circle.

Outline of the argument

Manuscript 1 (family 153, 2026-10-03): Measure concentration supplies finite bit-word differences whose initial coefficients can be completed to height-one polynomials with roots just above the parameter. Uniform finite-order nonvanishing retains the sign change, and irreducibility over the two-element field is imposed without changing the initial data, converting overlap concentration into one-sided algebraic-unit approximation.

Formal verification

Family 153 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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