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REVIEW 2 major objections 5 minor 34 references

On the Fourier transform of random Bernoulli convolutions

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For random Bernoulli convolutions, the geometric mean contraction exceeding 2/π puts the Fourier transform in L^1 almost surely, forcing a continuous density and interior support.

desk verdict A real improvement on random Bernoulli convolutions; the 2/π threshold is new and mostly well proved, but one Riemann-sum step at a singular endpoint needs a fix. read the letter →

arxiv 2507.21605 v3 pith:JGVCINFT submitted 2025-07-29 math.DS math.PR

Simon Baker , Henna Koivusalo , Sascha Troscheit , Xintian Zhang This is my paper · ORCID
classification math.DSmath.PR MSC 28A8042A3860G57
keywords randomBernoulliconvolutionsFouriertransformL1integrabilityabsolutecontinuityself-similarsetsRajchmanmeasurespolynomialdecayiteratedfunctionsystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies random Bernoulli convolutions, the distributions of infinite sums $\sum a_j \prod_{k=1}^j \lambda_k$ with $a_j\in\{0,1\}$ fair coins and with contraction ratios $\lambda_k$ i.i.d. uniform on a fixed interval $W\subset(0,1)$. Its central claim is that when the geometric mean contraction $\lambda_g=\exp\mathbb{E}(\log\lambda_1)$ exceeds $2/\pi$, the Fourier transform of the random measure is in $L^1(\mathbb R)$ almost surely. From there the measure is automatically absolutely continuous with a continuous density, and the random self-similar set supporting it has non-empty interior almost surely. This lowers the previously known threshold for interior points, about $0.824$, to about $0.636$, by estimating the $L^1$ norm of the Fourier transform directly. The paper also proves, without any restriction on $\lambda_g$, an almost-sure uniform polynomial decay rate for the Fourier transform.

What carries the argument

The engine is the infinite-product identity \[ |\widehat\mu_\omega(\xi)|=\prod_{j=1}^\infty\left|\cos\left(\pi\xi\prod_{k=1}^j\lambda_k\right)\right|, \] which converts the $L^1$ question into a quantitative question about how often the randomly scaled products $\xi\prod_{k=1}^j\lambda_k$ land near integer multiples of $\pi$. The proof partitions frequencies into geometric shells $I_i=[\lambda_g^{-i},\lambda_g^{-i-1})$, truncates the product at $E_i=\lfloor(1-\varepsilon)i\rfloor$, and conditions on the past through a martingale filtration. The key estimate (Lemma 9) bounds the conditional probability that the next cosine factor is as small as $\lambda_g^{h/M}$ by $(1+3\lambda_g^{\varepsilon^2 i}/\Delta)$ times the arccos-interval length divided by $\pi/2$. Summing these bounds yields a geometric series whose ratio is governed by the integral $\int_0^1 u/\sqrt{1-u^2}\,du=1$, and $\lambda_g>2/\pi$ is exactly the condition that makes the ratio less than $1$.

What would settle it

Simulate many realizations with $W$ chosen so that $\lambda_g$ is just above $2/\pi$, and for each realization numerically approximate the integral of the truncated product $\prod_{j=1}^N |\cos(\pi\xi\prod_{k=1}^j \lambda_k)|$ over growing frequency windows; if these integrals diverge as $N$ grows for a positive-measure set of $\omega$, Theorem 1 is false, whereas bounded growth on typical draws is the theorem's prediction.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the randomness of the contraction ratios is strong enough to overcome the algebraic obstructions that can make deterministic Bernoulli convolutions singular. For almost every $\omega$, whenever $\lambda_g > 2/\pi$ the Fourier transform satisfies $\widehat\mu_\omega \in L^1(\mathbb R)$, which by the classical criterion means $\mu_\omega$ is absolutely continuous with a continuous density and hence $\Lambda_\omega$ has non-empty interior. The proof actually gives more: there is a universal exponent $\rho>0$ such that almost surely $|\widehat\mu_\omega(\xi)|\le C_\omega|\xi|^{-\rho}$ for every $\xi\neq0$, with no hypothesis on $\lambda_g$ except the standing assumption $\lambda_{\max}<1$.

Load-bearing premise

The load-bearing premise is that each contraction ratio $\lambda_k$ is chosen randomly with a continuous range of possible values, uniformly on a fixed interval, so that the conditional probability of landing in any small target set is proportional to its length; if the $\lambda_k$ were fixed or atomic, the analogous conclusion can fail even when $\lambda_g>2/\pi$.

Editorial extensions

If this is right

  • If $\lambda_g > 2/\pi$, then for almost every contraction sequence the measure $\mu_\omega$ has an $L^1$ Fourier transform, hence a continuous density, and the random self-similar set $\Lambda_\omega$ has non-empty interior almost surely.
  • The interior-point threshold drops from the previous $e^{1/2}/2 \approx 0.824$ to $2/\pi \approx 0.636$, widening the parameter range known to produce absolutely continuous random Bernoulli convolutions.
  • Without any restriction on $\lambda_g$ (while $\lambda_{\max}<1$), there is a single exponent $\rho>0$ such that almost surely $|\widehat\mu_\omega(\xi)| \le C_\omega |\xi|^{-\rho}$ for all $\xi\neq0$; in particular, the measures are almost surely Rajchman with a uniform polynomial decay rate.
  • The paper does not settle whether $2/\pi$ is the sharp threshold, so the optimal boundary remains an open question inside $(1/2, 2/\pi]$, since $\lambda_g \le 1/2$ gives singular measures almost surely.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • For other atomless distributions on the contraction ratios, the same conditioning scheme should produce a threshold tied to how concentrated the distribution is; the value $2/\pi$ is a feature of the uniform model, not a universal constant for random self-similar sums.
  • Because the conditioning estimate uses the non-atomic law of each $\lambda_k$, the theorem cannot be transplanted to deterministic contraction sequences by continuity; known singular examples with rigid algebraic parameters show the analogous conclusion can fail in the deterministic setting.
  • A natural numerical probe is to let $\lambda_g = 2/\pi$ and watch the truncated $L^1$ integrals used in the proof: divergence there would indicate the constant is sharp, while convergence just below the boundary would suggest the true threshold is lower.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies random Bernoulli convolutions μ_ω formed as infinite convolutions with i.i.d. contraction ratios λ_k uniformly distributed on a fixed interval W ⊂ (0,1). Its main result, Theorem 1, states that if λ_g = exp(E log λ_1) > 2/π, then the Fourier transform of μ_ω is in L^1(R) almost surely. Theorem 2 then deduces that μ_ω is absolutely continuous with a continuous density and that its supporting random self-similar set has non-empty interior almost surely. Theorem 3 states that, without any lower bound on λ_g, the Fourier transform decays polynomially almost surely. The proof of Theorem 1 uses the infinite-product representation of |μ̂_ω|, a decomposition of frequencies into dyadic-like intervals I_i, a conditioning estimate for the 'good contraction' events (Lemma 9), and a limiting Riemann-sum computation that yields the constant 2/π. The paper thereby improves the earlier threshold e^{1/2}/2 ≈ 0.824 of Peres, Simon and Solomyak to 2/π ≈ 0.636.

Significance. If the proof is correct, this is a clean and substantial improvement over the previous threshold, and the exact constant 2/π is derived from an explicit integral rather than from any fitted parameter. The paper is self-contained, its structural reductions (Lemmas 4–8) are transparent, and Theorem 3 gives a useful almost-sure polynomial decay result that is new in this random setting. The uniform nature of the contraction-ratio law is genuinely used and is explicitly stated, which is appropriate. The main concern is one local but load-bearing limiting step in the proof of Theorem 1; once repaired, the paper would be a solid contribution to the Fourier analysis of random self-similar measures.

major comments (2)
  1. [§2, proof of Theorem 1, after Eq. (2.12)] The limiting identity that produces the constant 1, and hence the threshold 2/π, is not justified as written. The paper replaces the sum ∑_{k=0}^∞ λ_g^{k/M} (arccos λ_g^{(k+1)/M} − arccos λ_g^{k/M}) by the Riemann sum (1/M)∑_{k=0}^∞ λ_g^{k/M} f'(k/M), with f(x) = arccos(λ_g^x), and then by the integral ∫_0^∞ λ_g^x f'(x) dx. The difficulty is that f'(x) = −λ_g^x ln λ_g / √(1 − λ_g^{2x}) has a square-root singularity at x = 0. Consequently the term k = 0 in the displayed Riemann sum is infinite for every finite M, while the corresponding term f(1/M) − f(0) in the original sum tends to 0 as M → ∞. Thus the displayed equality between the original sum and that Riemann sum is not valid, and the sentence 'it immediately follows' does not constitute a proof. The limiting identity itself appears to be correct: it can be established by splitting off a neighbourhood of 0, estimating the first interval separately with f(1/M) − f(0) ≈ C/√M, and applying dominated convergence to the tail. Because this limit is what converts the exponent i/(E_i−n) into the numerical condition λ_g > 2/π, this gap is load-bearing and should be repaired in a revision.
  2. [§2, Lemma 9 and Remark 1] The paper's main theorem depends essentially on the conditional law of each λ_k given the past being absolutely continuous with a bounded density (here, uniform on W). This is exactly what makes the hitting probability of a small target interval proportional to its length in Lemma 9. The authors do state the uniform assumption, but Remark 1 generalizes only to 'another probability absolutely continuous with respect to Lebesgue'. I recommend making explicit that the argument does not extend to deterministic or atomic choices of λ_k, and that the threshold 2/π is tied to the non-atomic nature of the law; otherwise readers may over-read the result as depending only on λ_g. This is a clarification of the scope of the theorem rather than a correction of the proof.
minor comments (5)
  1. [§2, proof of Theorem 1, after Eq. (2.12)] The notation 'ε2i' appears in several places (e.g., in the statement of Lemma 9 and in the displayed estimates) and should be written as 'ε^2 i' to avoid ambiguity with 2ε i.
  2. [§2, proof of Theorem 1, after Eq. (2.12)] The sentence 'the second derivative of f'' is negative' should read 'f'' is negative'; also, the sign of f' is positive only because ln λ_g < 0, which could be stated explicitly.
  3. [§2, Eq. (2.12)] The infinite product in the definition of C_n is written with index i in one place and j in another; the index should be consistent.
  4. [§3, Lemma 13] The phrase 'for all large i large enough' should be 'for all sufficiently large i'.
  5. [§3, proof of Lemma 13] In the passage following Eq. (3.16), the statement that the right-hand side tends to infinity as p → 1 is correct, but the coefficient (ε − 1 + p/2) is negative for ε < 1/2; a brief explanation of why its product with log(1 − p) tends to +∞ would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 2/π threshold is computed from an explicit integral and the derivation is self-contained.

full rationale

This is a self-contained proof paper with no fitted parameters, no data, and no input that already contains the target conclusion. The threshold λ_g > 2/π is not assumed anywhere; it emerges from an explicit computation in the proof of Theorem 1. After Eq. (2.12), the authors show that the relevant geometric series converges to the Riemann-sum limit of the integral ∫_0^1 u/√(1-u^2) du = 1 (via the substitution u = λ_g^x), and they then verify that λ_g^{i/(E_i-n)} > 2/π follows from λ_g > 2/π for small ε and large i. Thus the constant 2/π is derived from the assumptions, not presupposed. Lemma 9's key hitting-probability estimate is proved in-text from the genuine assumption that each λ_k is uniform on a fixed interval W; the bound (arccos λ_g^{(h_k+1)/M} − arccos λ_g^{h_k/M})/(π/2) is obtained by direct interval counting, not imported. The only cited external facts are standard (the product formula for the Fourier transform, [Mat15, Theorem 3.4] for L^1 Fourier transforms implying absolute continuity), and these are appropriately attributed. Self-citations ([Koi14], [Tro17], [BB25], [BS23], [BKS24]) appear only as context or motivation for random self-similar measures and Fourier decay, and none is load-bearing for Theorem 1, Theorem 2, or Theorem 3. The skeptical concern that the Riemann-sum limit at the singular endpoint x = 0 of f(x) = arccos(λ_g^x) is not justified by the standard continuous-integrand theorem is a possible proof gap — the displayed identity is very likely correct and repairable by splitting off a neighborhood of 0 — but a gap in justifying a limit is a correctness risk, not circularity, since the limit's value (1) is computed, not assumed. No step in the derivation reduces to its own input by construction, and no load-bearing claim rests on the authors' own prior work rather than on the in-text argument.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests only on the explicit i.i.d. uniform model and standard probabilistic tools; no parameters are fitted to data and no ad hoc entities are introduced.

assumptions (6)
  • domain assumption The sequence (λ_k) is i.i.d. with uniform distribution on W = [λ_min, λ_max] ⊂ (0,1), with Δ = λ_max - λ_min > 0.
    Defines the random model; the absolute continuity and positive width of the law are used in Lemma 9 and Lemma 11 to estimate conditional hitting probabilities by interval lengths. This assumption is load-bearing: deterministic or atomic parameters can make the conclusion false (Pisot examples).
  • standard math Strong law of large numbers for the i.i.d. sequence (log λ_k): for each ε>0, the set B_ε(n) of sequences whose products stay within λ_g^{(1±ε)m} for all m ≥ n has probability tending to 1 as n→∞.
    Used in Lemmas 5, 6, 9 and 12 to restrict to 'good' contraction histories; the law of large numbers gives P(∪_n B_ε(n))=1.
  • standard math Fubini's theorem can be used to exchange the frequency integral and the expectation over ω.
    Used in Lemma 5 to pass from averaged L^1 integrability to almost-sure L^1 integrability.
  • standard math Borel-Cantelli lemma and the Chernoff bound for binomial tails.
    Used in Lemma 12 to prove that, almost surely, the products frequently land away from Zπ, giving polynomial decay.
  • standard math If a probability measure has Fourier transform in L^1(R), then it is absolutely continuous with a continuous density.
    Used to derive Theorem 2 from Theorem 1, citing Mattila [Mat15, Theorem 3.4].
  • standard math The Fourier transform of μ_ω is Lipschitz continuous with a uniform constant H because supp(μ_ω) ⊂ [0,(1-λ_max)^{-1}].
    Used in the proof of Theorem 3 to extend polynomial decay from the countable dense set of frequencies to all frequencies via the Lipschitz property.

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Pith. "Pith review of On the Fourier transform of random Bernoulli convolutions." pith.science (2026). https://pith.science/paper/JGVCINFT

@misc{pith2026250721605,
  author       = {Pith},
  title        = {Pith review of: On the Fourier transform of random Bernoulli convolutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JGVCINFT}},
  note         = {Machine review of arXiv:2507.21605}
}
abstract

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution \[ \mu_\omega = \mathop{\circledast}_{k=1}^{\infty} \left( \frac{\delta_0 + \delta_{\lambda_1 \lambda_2 \ldots \lambda_{k-1} \lambda_k}}{2} \right), \] where $\omega=(\lambda_k)$ is a sequence of i.i.d. random variables each following the uniform distribution on some fixed interval. We study the regularity of these measures and prove that when $\exp\mathbb{E}\left( \log \lambda_1\right)>\frac{2}{\pi}, $ the Fourier transform $\widehat{\mu}_\omega$ is an $L^{1}$ function almost surely. This in turn implies that the corresponding random self-similar set supporting $\mu_{\omega}$ has non-empty interior almost surely. This improves upon a previous bound due to Peres, Simon and Solomyak. Furthermore, under no assumptions on the value of $\exp \mathbb{E}(\log \lambda_1), $ we prove that $\widehat \mu_\omega$ will decay to zero at a polynomial rate almost surely.

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