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Absolute Continuity Parameters for Bernoulli Convolutions

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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 $\lambda\in(0,1)$, let $\mu_\lambda$ be the distribution of $\sum_{n\geq0}A_n\lambda^n$, where the $A_n$ are independent unbiased random variables taking values $\pm1$. Determine the set of parameters $\lambda$ for which $\mu_\lambda$ is absolutely continuous.

Formulation references: [S1], [S2].

Short English statement

For $\lambda\in(0,1)$, let $\mu_\lambda$ be the distribution of $\sum_{n\geq0}A_n\lambda^n$, where the $A_n$ are independent unbiased random variables taking values $\pm1$. Determine the set of parameters $\lambda$ for which $\mu_\lambda$ is absolutely continuous.

Sources

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

Contributions 1

Math-2424-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2424

Manuscript 1 (family 153, 2026-10-03): The independently defined finite algebraic-unit sets specify an explicit limsup approximation set C; the classification theorem states singular iff lambda belongs to C and absolute continuity iff not, covering every lambda in (0,1). A finite decision algorithm is not required by this record.

Absolute Continuity Parameters for 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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