Random Bernoulli Convolutions

• Random Bernoulli convolutions are probability measures constructed from infinite series with randomized contraction ratios, forming self-similar attractors.
• Their Fourier transform exhibits a Riesz product structure, enabling explicit criteria such as the λ₍g₎ > 2/π threshold for L¹ integrability.
• Parameter regimes ensure that fluctuations in the contraction ratios smooth the measure, leading to absolute continuity, continuous density, and non-empty support interiors.

Random Bernoulli convolutions are probability measures generated by infinite random series in which the contraction ratio is itself randomized from scale to scale. In the model studied in "On the Fourier transform of random Bernoulli convolutions" (Baker et al., 29 Jul 2025), one fixes a closed interval $W=[\lambda_{\min},\lambda_{\max}]\subset (0,1)$, lets $\omega=(\lambda_k)_{k\ge 1}$ be an i.i.d. sequence with each $\lambda_k$ uniformly distributed on $W$, and defines

$$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$

This random measure is the law of a Bernoulli series with random geometric weights, and it is supported on a random self-similar attractor $\Lambda_\omega$. The subject lies at the intersection of harmonic analysis, random iterated function systems, and the regularity theory of self-similar measures. Its recent development shows that randomization of contraction ratios can force substantial Fourier decay and, above an explicit threshold, absolute continuity with continuous density and non-empty interior of the support almost surely (Baker et al., 29 Jul 2025).

1. Definition and equivalent formulations

The basic random model is encoded by the geometric expectation

$\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$

where $\lambda_1\sim \mathrm{Unif}(W)$. The associated random Bernoulli convolution can be written in several equivalent ways. First, it is the infinite convolution already displayed above. Second, if $\nu$ denotes the Bernoulli $(1/2,1/2)$ product measure on $\omega=(\lambda_k)_{k\ge 1}$0, then $\omega=(\lambda_k)_{k\ge 1}$1, where

$\omega=(\lambda_k)_{k\ge 1}$2

Third, $\omega=(\lambda_k)_{k\ge 1}$3 is the law of the random series

$\omega=(\lambda_k)_{k\ge 1}$4

with $\omega=(\lambda_k)_{k\ge 1}$5 i.i.d. Bernoulli$\omega=(\lambda_k)_{k\ge 1}$6 (Baker et al., 29 Jul 2025).

These formulations emphasize different aspects of the model. The coding-map description identifies $\omega=(\lambda_k)_{k\ge 1}$7 as a random pushforward of a symbolic Bernoulli measure, while the series representation makes the analogy with classical deterministic Bernoulli convolutions immediate. The key distinction from the deterministic setting is that the scale $\omega=(\lambda_k)_{k\ge 1}$8 is replaced by the random product $\omega=(\lambda_k)_{k\ge 1}$9, so the contraction profile fluctuates from level to level.

The support is the random attractor

$\lambda_k$0

and $\lambda_k$1 for every $\lambda_k$2. Since $\lambda_k$3, the defining series converges and $\lambda_k$4 (Baker et al., 29 Jul 2025). The same work remarks that the arguments extend with minor technical changes to cases with $\lambda_k$5 provided $\lambda_k$6, which still guarantees $\lambda_k$7.

2. Support geometry and parameter regimes

Two elementary regimes are rigid. When $\lambda_k$8, the attractor $\lambda_k$9 is a Cantor set and $W$0 is singular for every $W$1. When $W$2, Borel–Cantelli implies that $W$3 has Lebesgue measure $W$4 almost surely, so $W$5 is singular almost surely (Baker et al., 29 Jul 2025). Thus the regime of interest begins when $W$6 meets $W$7.

The parameter $W$8 is the natural scale statistic because the products $W$9 track $$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$0 on high-probability events by the law of large numbers. For $$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$1 with uniform law,

$$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$2

hence

$$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$3

For example, if $$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$4 and $$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$5, then $$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$6, so the strongest regularity criterion presently available applies (Baker et al., 29 Jul 2025).

This geometry differs sharply from the classical one-parameter Bernoulli convolution $$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$7, the law of $$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$8 with fixed $$\mu_\omega=\mathop{\circledast}_{k=1}^{\infty}\left(\frac{\delta_0+\delta_{\lambda_1\lambda_2\cdots \lambda_k}}{2}\right).$$9 and i.i.d. digits $\Lambda_\omega$0. In that deterministic setting the support is always the interval $\Lambda_\omega$1, and multifractal phenomena can persist even in parameter regions where the measure is typically absolutely continuous (Jordan et al., 2010). The random model replaces a fixed overlap pattern by a fluctuating one, and the recent results indicate that this fluctuation has a smoothing effect (Baker et al., 29 Jul 2025).

3. Fourier transform, regularity, and interior

With the convention

$\Lambda_\omega$2

the Fourier transform of $\Lambda_\omega$3 has the random Riesz product form

$\Lambda_\omega$4

and therefore

$\Lambda_\omega$5

This identity reduces the regularity problem to estimating how often the random angles $\Lambda_\omega$6 fall near multiples of $\Lambda_\omega$7 (Baker et al., 29 Jul 2025).

The principal regularity theorem states that if $\Lambda_\omega$8, then $\Lambda_\omega$9 almost surely. Fourier inversion then yields an absolutely continuous measure with bounded continuous density, and because a nonnegative continuous density of total mass $\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$0 is strictly positive on some nonempty open interval, the support $\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$1 has non-empty interior almost surely (Baker et al., 29 Jul 2025). In the same regime, $\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$2 has Hausdorff dimension $\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$3.

A second theorem is unconditional: without any assumption on $\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$4, there exists $\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$5 such that for almost every $\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$6 there is $\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$7 with

$\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$8

The exponent can be chosen uniform in $\lambda_g:=\exp(\mathbb{E}(\log \lambda_1)),$9, although it depends on parameters selected in the argument, in particular on $\lambda_1\sim \mathrm{Unif}(W)$0 through auxiliary choices; no logarithmic corrections are required (Baker et al., 29 Jul 2025).

The new $\lambda_1\sim \mathrm{Unif}(W)$1 threshold improves earlier continuity and interior criteria summarized in the same source.

Condition on $\lambda_1\sim \mathrm{Unif}(W)$2	Consequence	Citation
$\lambda_1\sim \mathrm{Unif}(W)$3	$\lambda_1\sim \mathrm{Unif}(W)$4 with $\lambda_1\sim \mathrm{Unif}(W)$5 density a.s.	summarized in (Baker et al., 29 Jul 2025)
$\lambda_1\sim \mathrm{Unif}(W)$6	continuous density and non-empty interior a.s.	summarized in (Baker et al., 29 Jul 2025)
$\lambda_1\sim \mathrm{Unif}(W)$7	$\lambda_1\sim \mathrm{Unif}(W)$8, hence continuous density and interior a.s.	(Baker et al., 29 Jul 2025)

The interval $\lambda_1\sim \mathrm{Unif}(W)$9 remains structurally important. Absolute continuity may still hold there by other methods, but the $\nu$0 criterion does not guarantee it (Baker et al., 29 Jul 2025).

4. Proof architecture and the origin of the $\nu$1 threshold

The $\nu$2 argument begins by localizing to “good-product” events on which the random products $\nu$3 stay within a small multiplicative window around $\nu$4. For $\nu$5,

$\nu$6

and $\nu$7 as $\nu$8 by the law of large numbers (Baker et al., 29 Jul 2025). Frequency space is then partitioned into blocks $\nu$9, so that for $(1/2,1/2)$0 only the first $(1/2,1/2)$1 cosine factors are numerically relevant.

The next step is angular binning. For fixed $(1/2,1/2)$2, the proof partitions $(1/2,1/2)$3 into arcs

$(1/2,1/2)$4

and considers the event that $(1/2,1/2)$5 lands in $(1/2,1/2)$6. On $(1/2,1/2)$7, such membership forces a quantitative bound on the corresponding cosine factor. A filtration argument then gives conditional probability estimates of the form

$(1/2,1/2)$8

where $(1/2,1/2)$9 (Baker et al., 29 Jul 2025).

The denominator $\omega=(\lambda_k)_{k\ge 1}$00 is decisive. After summing over angular bins, integrating over $\omega=(\lambda_k)_{k\ge 1}$01, and passing $\omega=(\lambda_k)_{k\ge 1}$02, the discrete sum becomes an integral involving $\omega=(\lambda_k)_{k\ge 1}$03. The computation yields exactly

$\omega=(\lambda_k)_{k\ge 1}$04

after a change of variables, while the normalization contributes $\omega=(\lambda_k)_{k\ge 1}$05. Summability of the resulting series occurs precisely when the effective ratio beats $\omega=(\lambda_k)_{k\ge 1}$06, which is the source of the threshold $\omega=(\lambda_k)_{k\ge 1}$07 (Baker et al., 29 Jul 2025).

The polynomial-decay theorem uses a different but related mechanism. One discretizes each frequency block $\omega=(\lambda_k)_{k\ge 1}$08 by a net $\omega=(\lambda_k)_{k\ge 1}$09, proves that for a positive proportion of levels $\omega=(\lambda_k)_{k\ge 1}$10 the angles $\omega=(\lambda_k)_{k\ge 1}$11 fall in a fixed central strip $\omega=(\lambda_k)_{k\ge 1}$12, and then obtains

$\omega=(\lambda_k)_{k\ge 1}$13

The proportion estimate follows from conditional probability bounds and a Chernoff inequality for binomial tails; Borel–Cantelli then gives almost-sure control on the whole discretized net, and Lipschitz continuity of $\omega=(\lambda_k)_{k\ge 1}$14 transfers the bound to all frequencies (Baker et al., 29 Jul 2025). This is an Erdős–Kahane type discretization scheme in a genuinely random-contraction setting.

5. Relation to the classical Bernoulli convolution literature

Classical Bernoulli convolutions correspond to fixed contraction ratio. In the unbiased $\omega=(\lambda_k)_{k\ge 1}$15 normalization, $\omega=(\lambda_k)_{k\ge 1}$16 is the law of

$\omega=(\lambda_k)_{k\ge 1}$17

supported on $\omega=(\lambda_k)_{k\ge 1}$18, with Fourier transform

$\omega=(\lambda_k)_{k\ge 1}$19

under the $\omega=(\lambda_k)_{k\ge 1}$20-normalization (Varjú, 2016). For $\omega=(\lambda_k)_{k\ge 1}$21 the measure is singular and carried by a Cantor-type set; for $\omega=(\lambda_k)_{k\ge 1}$22 one enters the overlap regime, where a substantial body of work addresses dimension, entropy, absolute continuity, and exceptional algebraic parameters (Varjú, 2016).

Several deterministic benchmarks sharpen the contrast with the random model. For algebraic $\omega=(\lambda_k)_{k\ge 1}$23, the Hausdorff dimension satisfies

$\omega=(\lambda_k)_{k\ge 1}$24

where $\omega=(\lambda_k)_{k\ge 1}$25 is the asymptotic random-walk entropy; moreover $\omega=(\lambda_k)_{k\ge 1}$26 admits quantitative lower bounds in terms of the Mahler measure $\omega=(\lambda_k)_{k\ge 1}$27 (Breuillard et al., 2015). For transcendental parameters, $\omega=(\lambda_k)_{k\ge 1}$28 for every $\omega=(\lambda_k)_{k\ge 1}$29 (Varjú, 2018). Independently, a computer-assisted diffusion-operator method gives the uniform lower bound

$\omega=(\lambda_k)_{k\ge 1}$30

(Kleptsyn et al., 2021). These results concern dimension rather than $\omega=(\lambda_k)_{k\ge 1}$31-Fourier integrability, but they frame the deterministic landscape against which the random model should be read.

The deterministic theory also contains genuinely multifractal phenomena. For biased measures $\omega=(\lambda_k)_{k\ge 1}$32, the level sets

$\omega=(\lambda_k)_{k\ge 1}$33

can be nonempty and of positive Hausdorff dimension for many $\omega=(\lambda_k)_{k\ge 1}$34, including parameter regions where $\omega=(\lambda_k)_{k\ge 1}$35 is typically absolutely continuous (Jordan et al., 2010). This suggests that absolute continuity and exceptional fine-scale structure are compatible. A plausible implication for the random setting is that almost-sure regularity of $\omega=(\lambda_k)_{k\ge 1}$36 does not preclude nontrivial local-dimension phenomena, although the supplied results for random contractions focus on Fourier decay and interior rather than multifractal spectra.

Higher-dimensional analogues further enlarge the context. For the deterministic $\omega=(\lambda_k)_{k\ge 1}$37-dimensional Bernoulli convolution associated to

$\omega=(\lambda_k)_{k\ge 1}$38

one has

$\omega=(\lambda_k)_{k\ge 1}$39

under the condition that each $\omega=(\lambda_k)_{k\ge 1}$40 is not a root of a nonzero polynomial with coefficients in $\omega=(\lambda_k)_{k\ge 1}$41 (Rapaport et al., 2024). The random-contraction paper explicitly suggests possible extensions to higher-dimensional and random self-affine settings (Baker et al., 29 Jul 2025).

6. Generalizations, limitations, and open directions

The uniform law on $\omega=(\lambda_k)_{k\ge 1}$42 is mainly a technically transparent case. The proofs rely on absolute continuity of the law of $\omega=(\lambda_k)_{k\ge 1}$43, because geometric constraints on the angles $\omega=(\lambda_k)_{k\ge 1}$44 are converted into interval-length estimates. The same source remarks that analogous arguments work when $\omega=(\lambda_k)_{k\ge 1}$45 has a density on $\omega=(\lambda_k)_{k\ge 1}$46, but the quantitative constants, including the threshold replacing $\omega=(\lambda_k)_{k\ge 1}$47, change with the law; in the uniform case the explicit constant $\omega=(\lambda_k)_{k\ge 1}$48 emerges from arccos geometry together with equal weighting across $\omega=(\lambda_k)_{k\ge 1}$49 (Baker et al., 29 Jul 2025).

Three open problems are singled out. First, the optimality of the $\omega=(\lambda_k)_{k\ge 1}$50 threshold is unknown. Lowering it, or proving that it is best possible for $\omega=(\lambda_k)_{k\ge 1}$51-Fourier integrability and interior, is a natural problem (Baker et al., 29 Jul 2025). Second, the distinction between positive Lebesgue measure and non-empty interior remains delicate: in the deterministic setting, whether a self-similar set in $\omega=(\lambda_k)_{k\ge 1}$52 can have positive Lebesgue measure but empty interior is still open, and random models may clarify the typical geometry (Baker et al., 29 Jul 2025). Third, a systematic theory for non-uniform contraction laws, and beyond that for random graph-directed or self-affine systems, is still undeveloped (Baker et al., 29 Jul 2025).

Taken together, the available results place random Bernoulli convolutions in a distinct position within the broader Bernoulli-convolution literature. The model retains the Riesz-product Fourier structure of the classical case, but randomizes the contraction geometry strongly enough to yield almost-sure polynomial Fourier decay without any assumption on $\omega=(\lambda_k)_{k\ge 1}$53, and $\omega=(\lambda_k)_{k\ge 1}$54-Fourier integrability, continuous density, and non-empty interior once $\omega=(\lambda_k)_{k\ge 1}$55 (Baker et al., 29 Jul 2025). In that sense, the theory identifies a concrete harmonic-analytic mechanism by which random scale fluctuations regularize a family of highly overlapping self-similar measures.