Bernoulli convolutions: measure type and non-Pisot singularity

Source and verification

This article explains family 153 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Bernoulli convolutions and measure type

For $0<\lambda <1$, let

$$\begin{array}{c}{Y}_{\lambda }=\sum _{n=0}^{\infty }{\epsilon }_n{\lambda }^n,\\ \mathbb{P}({\epsilon }_n=\pm 1)=\frac{1}{2},\end{array}$$

with independent signs. Its law is the unbiased Bernoulli convolution ${\nu }_{\lambda }$. Absolute convergence is immediate.

The bit version $X_{\lambda }={\sum }_{n\ge 0}{u}_n{\lambda }^n$, with independent fair bits, satisfies

$$Y_{\lambda }=2X_{\lambda }-\frac{1}{1-\lambda }.$$

This affine change preserves absolute continuity and singularity. A singular measure is carried by a Lebesgue-null Borel set.

These measures have pure type: either singular or absolutely continuous. One way to see this is to split an invariant measure into its absolutely continuous and singular parts. The two affine branches preserve both types. If both parts had positive mass, normalizing them would give two distinct invariant probabilities, contradicting uniqueness of the contracting fixed-point law.

🏷️ Baseline parameters

For $\lambda <1/2$, conditioning on the first $n$ bits covers the support by $2^n$ intervals of length ${\lambda }^n/(1-\lambda )$. Their total length tends to zero. Hence the measure is singular.

At $\lambda =1/2$, binary expansion gives the uniform bit law on $[0,2]$, so ${\nu }_{1/2}$ is uniform on $[-2,2]$.

For $\lambda >1/2$, overlaps make measure type subtle. Full Hausdorff dimension does not imply absolute continuity. Likewise, decay of a Fourier transform is necessary for an integrable density but not sufficient to establish one.

The manuscript claims a criterion for every parameter and two classes of non-Pisot singular examples. (OpenAI, 2026)

🏷️ A finite arithmetic predicate and an infinite criterion

The criterion uses finite polynomial data; the exposition here records that data without developing a separate algebra course.

For integers $n\ge 1$ and $d\ge 2$, let $R(n,d)$ be the real algebraic units in $(1/2,1)$ of degree $dn$ whose minimal polynomials have coefficients in $\{-1,0,1\}$ and remain irreducible modulo two. Normalize the polynomial’s constant coefficient to $-1$.

For $r\in R(n,d)$, collect the first $n$ coefficients of the normalized polynomials of all $q\in R(n,d)$ with $|q-r|\le 4^{-n}$. Call this set of distinct vectors $W_n(r;d)$. Repeated coefficient vectors are counted once.

A bit word $u$ specifies a closed coordinate orthant by $(2u_i-1)w_i\ge 0$. Retain $r$ in $R_{\ast }(n,d,j)$ when at least $2^n/d$ of these orthants each contain at least $j(2r{)}^n$ vectors of $W_n(r;d)$.

Classification in the manuscript

For $\lambda \in (1/2,1)$, singularity is equivalent to the following condition: there exists one $d\ge 2$ such that, for every integer $j\ge 1$, there are infinitely many $n$ and retained roots $r\in R_{\ast }(n,d,j)$ satisfying

$$r-4^{-n}\le \lambda <r.$$

Replacing “infinitely many” by “every sufficiently large” gives the same parameter set. (OpenAI, 2026)

The degree multiplier $d$ is selected before the crowding level $j$. The threshold in $n$ may depend on $j$. The interval is one-sided and excludes $r$ itself.

Every finite target set can be enumerated through polynomial arithmetic. Membership in the resulting infinite approximation condition is not a finite decision algorithm for an arbitrary real parameter.

🏷️ Crowded prefixes force singularity

Write $s_u(\lambda )={\sum }_{i<n}{u}_i{\lambda }^i$. If a coefficient vector $w$ lies in the orthant of $u$, then $v=u-w$ is another bit word. Near-root estimates make $s_u(\lambda )$ and $s_v(\lambda )$ differ by at most a constant times ${\lambda }^n$.

The retained-root count therefore supplies many words with many nearby neighbors. Divide the line into bins of length ${\lambda }^n$. Counting word-bin incidences produces a union $E_j$ of slightly enlarged bins such that

$$\mathrm{Leb}(E_j)\le \frac{C_{\lambda }}{j},{\mu }_{\lambda }(E_j)\ge \frac{1}{d}.$$

An absolutely continuous finite measure gives uniformly small mass to sets of sufficiently small Lebesgue measure. Letting $j\to \infty$ rules out absolute continuity; pure type then gives singularity.

The converse is the harder construction. Concentration yields many close word pairs, whose coefficient differences must be completed into the prescribed polynomials with roots strictly above $\lambda$. A finite-order sign-change estimate and a polynomial completion argument preserve the required initial coefficients.

🏷️ Singular examples with decaying Fourier transform

A Pisot number is a real algebraic integer greater than one whose other conjugates lie strictly inside the unit circle. The classical reciprocal-Pisot singularity mechanism uses Fourier nondecay.

A quartic Salem number has conjugates $\beta ,{\beta }^{-1},z,\bar{z}$, with $\beta >1$ and $|z|=1$. The manuscript claims singularity of ${\nu }_{1/\beta }$ for every such $\beta \in (1,2)$. Its Fourier transform tends to zero, so the Pisot nondecay argument cannot establish this conclusion. (OpenAI, 2026)

The source constructs multipliers making the unit-circle conjugates small while controlling the accumulated cosine losses. Trace identities localize selected Fourier phases to nearly disjoint blocks of random signs. Averages of these phases concentrate under the Bernoulli law while tending to zero in Lebesgue mean square. The separating statistic uses many frequencies rather than one nondecaying Fourier sequence.

A second example uses the root $\beta \in (1.8392,1.8393)$ of

$$p(t)=t^{31}-2t^{30}+\sum _{j=0}^9(t^{3j+1}-t^{3j}).$$

The source claims that $\beta$ is non-Pisot and ${\nu }_{1/\beta }$ singular. A weakly expanding conjugate pair makes a crude lattice count too large. A product of positive trigonometric factors gives the additional typical-word saving needed to overcome that excess. Root enclosures and moment estimates use rational certificates, which were not executed for this article.

🏷️ Key steps and proof outlines

Crowding yields sets with small length and definite mass

At a retained root, the orthant condition associates to many bit words $u$ a large collection of coefficient differences $w$ for which $u-w$ is again a bit word. The near-root estimate puts their prefix sums within a fixed multiple of ${\lambda }^n$ of the sum for $u$. Thus the combinatorial condition is a multiplicity statement about actual nearby prefix values, not just nearby polynomial roots.

Partition the line into intervals of length ${\lambda }^n$. A crowded word lies near a bin containing order $j(2\lambda {)}^n$ word occurrences. Since there are $2^n$ total words, only order $2^n/[j(2\lambda {)}^n]=1/(j{\lambda }^n)$ such heavy bins are needed. Enlarging each by a fixed number of neighboring bins and by the tail interval changes only a parameter-dependent constant. Their total length is at most $C_{\lambda }/j$, while the prescribed positive fraction of crowded words gives mass at least $1/d$.

If the measure were absolutely continuous, uniform absolute continuity of its integral would make the mass of all sufficiently short measurable sets small, contradicting this fixed lower bound as $j\to \infty$. Pure type then implies singularity. The converse uses the source’s sign-change and polynomial-completion theorems to build admissible roots from concentration; it does not follow merely by reversing the bin count. (OpenAI, 2026)

Many Fourier phases can distinguish measures even when each decays

For the quartic Salem case, the source constructs continuous statistics

$$F_m(y)=\frac{1}{M_m}\sum _{n\in {\mathcal{N}}_m}{\sigma }_{m,n}{e}^{i\pi A_m{\beta }^{n}y},$$

where ${\sigma }_{m,n}\in \{-1,1\}$ aligns the real expectations. Its arithmetic multiplier construction gives a scale $L_m\to \infty$, a mean $a_m=\mathbb{E}{F}_m(Y)\ge e^{-L_m/20}$, and at least $M_m\ge e^{0.7L_m}$ selected frequencies.

Trace identities replace remote phase coefficients by integers. Their contributions to the exponential become deterministic signs. The remaining phase for each selected frequency depends, up to small $L^2$ error, on a finite block of Bernoulli signs. The selected blocks are disjoint, so these approximating phase variables are independent. Summing variances and then adding the approximation error gives

$$\Vert F_m(Y)-a_m{\Vert }_2\le M_m^{-1/2}+C{e}^{-L_m/4}.$$

Independence is asserted for those finite-block approximations, not for the original overlapping infinite Fourier phases. The scale choices make the error much smaller than the possibly vanishing mean $a_m$.

Lebesgue orthogonality supplies the opposing estimate

Order the frequencies as ${\xi }_1<\cdots <{\xi }_{M_m}$. Their separation satisfies $|{\xi }_r-{\xi }_s|\ge \pi (\beta -1)|r-s|$. On a fixed interval $I$ containing the support,

$$\left|{\int }_I{e}^{i({\xi }_r-{\xi }_s)y}dy\right|\le \frac{2}{|{\xi }_r-{\xi }_s|}(r\ne s).$$

Expanding the squared average and summing by $|r-s|$ yields ${\int }_I|F_m{|}^2\le C(1+\log M_m)/M_m$. No exact integer-frequency orthogonality is needed; this integral bound handles the nonintegral frequencies directly.

Define $E_m=\{y\in I:\Re F_m(y)\ge \frac{1}{2}{e}^{-L_m/20}\}$. Chebyshev’s inequality under the Bernoulli law gives $\nu (E_m)\to 1$, whereas the Lebesgue $L^2$ estimate gives

$$|E_m|\le C{e}^{L_m/10}\frac{1+\log M_m}{M_m}⟶0.$$

Choose a subsequence for which both $|E_m|$ and $1-\nu (E_m)$ are summable. Its limsup is Lebesgue null and carries full Bernoulli measure. This proves singularity through a family of increasingly sensitive tests even though individual Fourier values tend to zero.

The degree-thirty-one example uses a different quantitative saving

The separate non-Pisot construction has a weakly expanding conjugate pair, so its raw lattice count is insufficient. The source weights typical words by a product of positive trigonometric factors; a certified moment estimate supplies an exponential saving that offsets that extra expansion. Its polynomial root locations and strict numerical inequalities are finite rational certificates, distinct from the analytic many-frequency separation in the Salem argument. The source’s certificate status is therefore part of this conclusion; the present exposition has not executed those computations. Neither route infers singularity from full dimension or from Fourier decay alone.

See Also

Self-similar dimension gives an entropy-rate formula that does not determine absolute continuity. Fourier convergence provides complementary background on the difference between frequency estimates and pointwise measure-theoretic conclusions.

🏷️ Manuscripts and verification scope

• Arithmetic classification and non-Pisot singularity for Bernoulli convolutions — pinned manuscript and companion source.

No family scope page was present at lean/docs/153.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Arithmetic classification and non-Pisot singularity for Bernoulli convolutions.