When a random geometric series has a density

← All digests

Family 153 · Dynamical systems and ergodic theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

Toss a fair coin repeatedly, turn the outcomes into signs $\varepsilon_n=\pm1$, and form a convergent random series

$$Y_\lambda=\sum_{n=0}^{\infty}\varepsilon_n\lambda^n, \qquad 0<\lambda<1.$$

Its distribution $\nu_\lambda$ is an unbiased Bernoulli convolution. Does it have a density, or does all its probability hide on a set of ordinary length zero?

For small $\lambda$, the possible values form a thin Cantor-like set. When $\lambda>1/2$, the support is an interval, yet the distribution can still be singular. Overlap between different coin histories creates an arithmetic problem rather than a simple support question.

Family 153 claims a classification for every parameter through an explicitly defined approximation condition. It also claims singular distributions at reciprocals of quartic Salem numbers and at one specified non-Pisot algebraic parameter. The classification is an infinite criterion, not a finite decision algorithm.

2. Background and essential definitions

A measure is absolutely continuous with respect to Lebesgue measure if it has a density: probabilities can be obtained by integrating a nonnegative function. It is singular if some set of Lebesgue measure zero carries its entire mass. Bernoulli convolutions have pure type, so these alternatives exhaust the possibilities.

For $\lambda<1/2$, after fixing $n$ signs there are $2^n$ possible prefixes, each with a remaining tail contained in an interval of length proportional to $\lambda^n$. The total covering length tends to zero because $(2\lambda)^n\to0$. At $\lambda=1/2$, binary expansion instead gives the uniform distribution on $[-2,2]$. The difficult region is $(1/2,1)$.

It is often convenient to use fair bits $u_n\in\{0,1\}$. Their sum $X_\lambda=\sum u_n\lambda^n$ satisfies

$$Y_\lambda=2X_\lambda-\frac1{1-\lambda},$$

so the two laws have the same measure type. Differences between finite bit sums are values at $\lambda$ of polynomials with coefficients in $\{-1,0,1\}$. Exact or approximate polynomial relations therefore encode overlapping histories.

Algebraic conjugates and exceptional parameters

An algebraic integer is a root of a monic polynomial with integer coefficients. Its conjugates are the other roots of its minimal polynomial over the rationals. A Pisot number is a real algebraic integer $\beta>1$ whose other conjugates all have modulus less than one.

A Salem number is a real algebraic integer $\beta>1$ whose other conjugates are $\beta^{-1}$ and numbers on the complex unit circle. In degree four, there is one nonreal conjugate pair on that circle.

These definitions describe how one algebraic relation behaves under all its embeddings. That behavior controls how powers of $\beta$ approach integers and how Fourier phases of the random sum align.

Full Hausdorff dimension is a different property from absolute continuity. A measure can have dimension one and still live on a set of Lebesgue measure zero. This distinction is essential when comparing this family with the dimension formula in family 148.

3. Development history and previous partial results

Erdős proved in 1939 that Bernoulli convolutions are singular when $\lambda^{-1}$ is Pisot. His method shows that their Fourier transforms fail to tend to zero. An integrable density would force Fourier decay, so nondecay excludes absolute continuity.

Garsia subsequently established absolute continuity for a different arithmetic class, including reciprocals of algebraic integers of absolute norm two whose conjugates all lie outside the unit circle. He also connected entropy of finite sums with singularity. The subject thus acquired both arithmetic examples and information-theoretic tools.

Solomyak proved that for Lebesgue-almost every $\lambda\in(1/2,1)$, the distribution has an $L^2$ density. Hochman’s work connected dimension loss with extremely strong concentration of finite-level positions, and Shmerkin proved that the exceptional set for absolute continuity has Hausdorff dimension zero. These results make singular parameters very rare without identifying every one of them.

Varjú proved full dimension for every transcendental parameter in this interval. Breuillard and Varjú showed that dimension-deficient parameters admit very close algebraic approximations by other dimension-deficient parameters. Neither conclusion alone classifies absolute continuity at each exceptional parameter.

Salem’s Fourier result creates a further obstacle: when the reciprocal is not Pisot, the Fourier transform tends to zero. Thus Erdős’s nondecay argument cannot establish singularity at Salem parameters. Recent quantitative decay estimates for Salem convolutions likewise do not settle their measure type.

The new manuscript presents the existence of singular non-Pisot examples as an unresolved issue in the recent literature it cites. Its contribution is claimed at two levels: a universal arithmetic characterization and direct constructions of singular examples beyond the classical Pisot mechanism.

4. What the new manuscripts claim

The manuscript, dated 3 October 2026, has three principal claims.

A criterion made entirely from finite algebraic data

For integers $n\ge1$ and $d\ge2$, form a finite set $R(n,d)$ of real algebraic units in $(1/2,1)$. An algebraic unit has both itself and its inverse algebraic integral. Here its minimal polynomial must have degree $dn$, coefficients in $\{-1,0,1\}$, and irreducible reduction modulo two. Choose its sign so that the constant coefficient is $-1$.

For a candidate $r$, inspect other members $q$ of this set within distance $4^{-n}$. Record the first $n$ coefficients of their polynomials, counting distinct coefficient vectors only once. Call this collection $W_n(r;d)$.

Each bit word $u\in\{0,1\}^n$ specifies a closed coordinate orthant: coordinate $i$ is nonnegative when $u_i=1$, and nonpositive otherwise. Zero coordinates belong to both relevant sign choices. Retain $r$ at crowding level $j$ if at least $2^n/d$ of these orthants each contain at least

$$j(2r)^n$$

distinct vectors from $W_n(r;d)$.

The claimed criterion says that $\nu_\lambda$, for $\lambda\in(1/2,1)$, is singular exactly when there exists a fixed $d\ge2$ such that, for every integer $j\ge1$, retained targets $r$ satisfy

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

for arbitrarily large $n$. The manuscript also claims that “arbitrarily large” can be replaced by “every sufficiently large,” with the threshold depending on $j$.

The order of quantifiers matters: one degree multiplier $d$ works for all crowding levels. The target lies strictly above $\lambda$. Every finite target set uses polynomial arithmetic alone, without consulting which Bernoulli convolutions are singular. Parameters failing the criterion are absolutely continuous; the cases at and below $1/2$ are as described earlier.

Explicit non-Pisot singularity

The manuscript claims singularity for $\nu_{1/\beta}$ whenever $\beta\in(1,2)$ is a quartic Salem number. In particular, it gives the real roots greater than one of

$$x^4-x^3-x^2-x+1, \qquad x^4-2x^3+x^2-2x+1.$$

A separate claim concerns

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

The specified real root $\beta\in(1.8392,1.8393)$ is claimed to be non-Pisot, with $\nu_{1/\beta}$ singular. The polynomial has an additional conjugate pair just outside the unit circle, so this example uses a different configuration from quartic Salem numbers.

5. Why the advance matters

Provisional importance: 5/5. If correct, the family resolves a central measure-type classification question and supplies exceptional examples that the classical Fourier-nondecay mechanism cannot reach.

The universal criterion is valuable even though it is not a short membership test. It translates a statement about concentration of an infinite random series into a prescribed pattern of algebraic approximations and coefficient directions. That identifies what arithmetic data would be sufficient and necessary.

The non-Pisot examples have a complementary role. They would show concretely that Fourier decay does not settle absolute continuity in this classical family. The general criterion and the examples are separate contributions: the manuscript proves the examples directly, then obtains their approximation properties from the criterion.

6. The main idea

From crowded polynomials to concentrated probability

Suppose many coefficient vectors occur near a target root. A vector lying in the orthant specified by a bit word $u$ can be subtracted from $u$ to produce another bit word. Since the corresponding polynomial nearly vanishes at $\lambda$, these different words have nearby prefix sums.

If this happens in enough orthants, a positive fraction of the probability is covered by unusually few short intervals. Raising the crowding level makes the total interval length arbitrarily small while preserving a fixed positive mass. This rules out absolute continuity; pure type then gives singularity.

Conversely, concentration of a singular measure supplies many pairs of close finite sums with different first bits. Their difference polynomials have the desired nonzero constant coefficient. The manuscript completes their initial coefficients into polynomials with roots strictly above $\lambda$, while keeping the root error small.

Two tools have different jobs. A finite-order nonvanishing estimate maintains a sign change and hence a nearby root. Irreducible polynomials in arithmetic progressions over $\mathbb F_2$ provide the prescribed irreducibility without changing the initial coefficient pattern. The completed roots become the algebraic approximation targets.

Detect singularity despite Fourier decay

For quartic Salem parameters, algebraic trace identities relate the real expanding conjugate to its unit-circle pair. The construction chooses multipliers whose unit-circle contributions become very small while controlling accumulated cosine losses.

Rather than seek one Fourier coefficient that refuses to decay, the manuscript builds averages of many phases. Their relevant random-sign windows can be separated. The averages concentrate under the Bernoulli convolution but tend to zero in Lebesgue mean square. Sets detected by these averages can therefore carry nearly all the random-series mass while having arbitrarily small ordinary length.

For the degree-31 example, finite words are evaluated simultaneously at all roots, placing their values in a lattice. The extra expanding conjugate pair makes a basic lattice count slightly too large. A product of positive trigonometric factors supplies a correction: its Lebesgue integral stays bounded, while it grows exponentially at typical word evaluations. The resulting concentration gain outweighs the extra volume growth.

The manuscript supplies finite recurrences and rational error bounds for numerical constants used in this second construction. Their role here is part of the reported argument; this exposition does not rerun a mathematical verification.

7. Scope, limitations, and open questions

The series uses independent fair signs. Biased Bernoulli convolutions require different statements. The universal criterion covers all real $0<\lambda<1$, but involves infinitely many scales and crowding levels. Finite enumeration at each scale is not a finite procedure deciding membership for an arbitrary real input.

The Salem theorem concerns degree four and $1<\beta<2$. It does not assert singularity for every Salem number of arbitrary degree. The degree-31 construction likewise gives one specified parameter, not a general theorem for every non-Pisot algebraic reciprocal.

The criterion separates singularity from absolute continuity. It does not automatically provide an $L^2$, bounded, or differentiable density in every absolutely continuous case, nor an effective convergence rate for approximating that density.

Understanding how to use the criterion efficiently for concrete parameters, and how far the direct non-Pisot mechanisms extend, remain distinct quantitative and structural questions.

8. Sources and further reading

Erdős’s 1939 singularity theorem and Garsia’s 1962 arithmetic study establish the classical contrasting examples. Solomyak’s almost-everywhere density theorem describes the typical parameter.

Hochman’s entropy and overlap theorem, Shmerkin’s exceptional-set theorem, and Varjú’s transcendental-parameter dimension theorem explain why dimension and measure type must be distinguished. Salem’s Fourier analysis of exceptional sets supplies the classical decay context.

Source manuscripts

• M1. OpenAI, Arithmetic classification and non-Pisot singularity for Bernoulli convolutions (October 3, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.