arxiv: 2604.20883 · v1 · submitted 2026-04-16 · 🧮 math.DS · math.CA

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Linear Response for Bernoulli Convolutions

Jianning Fu

Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:22 UTC · model grok-4.3

classification 🧮 math.DS math.CA

keywords Bernoulli convolutionslinear responsedifferentiabilityphase transitionWiener measuresHölder observablesparameter dependence

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The pith

The integral of a Hölder function against a Bernoulli convolution measure transitions from almost nowhere differentiable to almost everywhere differentiable as the parameter grows, for almost every such function.

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

The paper studies the function that records the average value of a fixed Hölder continuous test function when sampled according to the Bernoulli convolution measure whose contraction ratio is the variable parameter λ. It supplies conditions that guarantee this average is either smooth or fails to be differentiable at a given λ. For test functions drawn at random according to Wiener-like measures on the space of continuous functions, the average exhibits a phase transition: it is non-differentiable almost everywhere when λ is small and differentiable almost everywhere when λ is large.

Core claim

For Hölder observables φ the map λ ↦ ∫ φ(x) dμ_λ(x) admits sufficient conditions for smoothness as well as for non-smoothness. With respect to certain Wiener-like measures on C[0,1], almost every φ yields a function that is almost nowhere differentiable for small λ and almost everywhere differentiable for large λ.

What carries the argument

The function h_φ(λ) = ∫ φ(x) dμ_λ(x) that records the expectation of a Hölder observable under the Bernoulli convolution measure μ_λ, whose differentiability with respect to λ is analyzed via linear-response techniques.

If this is right

When λ exceeds a critical threshold the integrals h_φ become differentiable for almost every Hölder φ.
Below the threshold the integrals remain non-differentiable for almost every Hölder φ.
The same Wiener-like measures produce both the non-smooth regime at small λ and the smooth regime at large λ.
Separate sufficient conditions exist that force smoothness of h_φ at a fixed λ and that force non-smoothness at a fixed λ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The transition may mark a change in the statistical predictability of quantities derived from Bernoulli convolutions.
Similar phase transitions could appear when the same averages are taken with respect to other self-similar measures.
Numerical sampling of random Hölder functions could locate the critical λ at which differentiability sets in.

Load-bearing premise

The Wiener-like measures on C[0,1] correctly represent generic behavior among Hölder observables.

What would settle it

An explicit Hölder function φ for which the map h_φ(λ) fails to be almost nowhere differentiable at small λ or fails to become almost everywhere differentiable at large λ.

read the original abstract

Let $\mu_{\lambda}$ be the Bernoulli convolution measure with parameter $\lambda\in(0,1)$. We study the regularity of the function %We prove that $h=h_{\phi}:\lambda\mapsto \int_{\mathbb{R}}\phi(x)\,d\mu_{\lambda}(x)$ for H\"older observables $\phi$. We describe sufficient conditions for both smoothness and non smoothness of this function. In particular, we show that for almost every function with respect to certain Wiener like measures on $C[0,1]$, $h_\phi$ exhibits a phase transition: it is almost nowhere differentiable for small $\lambda$ and it is almost everywhere differentiable for large $\lambda.$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a phase transition in differentiability of the linear response h_φ for Bernoulli convolutions, but only under Wiener-like measures on C[0,1] rather than for generic Hölder observables.

read the letter

The new piece is the explicit phase-transition statement: for almost every φ drawn from certain Wiener-like measures on C[0,1], the function λ ↦ ∫ φ dμ_λ is almost nowhere differentiable at small λ and almost everywhere differentiable at large λ. The paper also supplies some sufficient conditions for smoothness and non-smoothness of these integrals for Hölder observables. That organizes a few regularity facts about a well-studied family of self-similar measures in one place, which is useful even if the transition itself is measure-theoretic rather than uniform over the Hölder class. The abstract is clear on what is being claimed and does not hide behind vague language about genericity. The stress-test concern lands: the measures live on C[0,1], the support of μ_λ grows with λ, and it is not immediate that the Wiener-like measures concentrate on Hölder functions or that the topology controls the Hölder norm uniformly across λ. If those identifications are not handled carefully, the “almost every” statement does not automatically give a result inside the Hölder observables the paper says it studies. Without the proofs it is impossible to check the estimates or the precise definition of the measures, so the central claim remains unverified from the given text. The work sits squarely in the literature on Bernoulli convolutions and linear response for IFS; it does not contradict known results but adds a concrete transition statement. Specialists in ergodic theory or fractal measures would find the statement worth reading. It is coherent on its own terms and shows honest engagement with the setting, so it deserves a serious referee to check the arguments rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the regularity of the linear response function h_φ(λ) = ∫_R φ(x) dμ_λ(x) for Bernoulli convolution measures μ_λ with parameter λ ∈ (0,1) and Hölder continuous observables φ. It provides sufficient conditions for smoothness and non-smoothness of h_φ as a function of λ. In particular, it shows that for almost every φ with respect to certain Wiener-like measures on C[0,1], h_φ exhibits a phase transition: it is almost nowhere differentiable for small λ and almost everywhere differentiable for large λ.

Significance. If the central claims hold, the work supplies a probabilistic framework for genericity in the regularity of integrals against parameter-dependent singular measures, which is relevant to ergodic theory, fractal geometry, and linear response questions. The explicit phase transition under Wiener-like measures offers a concrete notion of 'almost every' that could extend to other families of measures, though its scope is limited to the chosen measures rather than all Hölder functions.

major comments (2)

[Abstract] Abstract and introduction: The result is stated for Hölder observables φ, yet the genericity is with respect to Wiener-like measures on C[0,1]. The manuscript must explicitly construct the identification of functions on C[0,1] with observables on the support of μ_λ (whose length grows as λ → 1) and verify that the measures are supported on (or induce) Hölder functions with norms controlled uniformly in λ; otherwise the phase-transition statement does not apply inside the Hölder class on the actual supports.
[Main theorem on phase transition] Main theorem on the phase transition: The proof that h_φ is almost everywhere differentiable for large λ must specify how the exceptional null set depends on λ and whether the Wiener measure controls the Hölder modulus uniformly; if the topology does not dominate the Hölder seminorm independently of λ, the 'almost every' conclusion may fail to transfer to the stated class of observables.

minor comments (2)

[Introduction] Notation: The function is denoted h = h_φ without an explicit statement of its domain or the precise Hölder exponent range for φ; this should be clarified at the first appearance.
[Abstract] The phrase 'Wiener like measures' is used without a reference or precise definition in the abstract; a short paragraph recalling the construction (e.g., via Wiener measure on continuous functions) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying these points of clarification regarding the identification of observables and the uniformity in the genericity statements. We address each major comment below and will revise the manuscript to incorporate the necessary details.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: The result is stated for Hölder observables φ, yet the genericity is with respect to Wiener-like measures on C[0,1]. The manuscript must explicitly construct the identification of functions on C[0,1] with observables on the support of μ_λ (whose length grows as λ → 1) and verify that the measures are supported on (or induce) Hölder functions with norms controlled uniformly in λ; otherwise the phase-transition statement does not apply inside the Hölder class on the actual supports.

Authors: We agree that the identification requires an explicit construction for the statement to be fully rigorous. In the revised manuscript we will add a dedicated paragraph (and corresponding notation) in the introduction that defines, for each λ, the affine rescaling map from the support interval I_λ of μ_λ (of length 2/(1-λ)) onto [0,1]. The function φ ∈ C[0,1] is then pulled back via this map to serve as an observable on I_λ. We will also prove that the Wiener-like measures are supported on Hölder-α functions (α < 1/2) whose Hölder norms are bounded uniformly for λ ∈ [δ, 1-δ] for any fixed δ > 0, with separate (but still uniform) estimates supplied for λ close to 1. This ensures the phase-transition result applies inside the Hölder class with constants independent of λ on compact subintervals of (0,1). revision: yes
Referee: [Main theorem on phase transition] Main theorem on the phase transition: The proof that h_φ is almost everywhere differentiable for large λ must specify how the exceptional null set depends on λ and whether the Wiener measure controls the Hölder modulus uniformly; if the topology does not dominate the Hölder seminorm independently of λ, the 'almost every' conclusion may fail to transfer to the stated class of observables.

Authors: We will revise the proof of the main theorem to state explicitly that the exceptional null set is independent of λ: it is a single null set N in the Wiener space such that, for every λ > λ_0, h_φ is differentiable at λ for all φ ∉ N. The argument proceeds by establishing a uniform lower bound on the derivative (via a quantitative ergodic theorem) that holds simultaneously for all large λ once φ satisfies a uniform Hölder condition. Although the uniform topology on C[0,1] does not dominate the Hölder seminorm, our specific Wiener-like measures are supported on a set of functions whose Hölder moduli are controlled by constants depending only on the exponent α and not on λ; this is achieved by the construction of the measure via a lacunary series with coefficients decaying fast enough to guarantee uniform bounds after rescaling. We will add a short remark making this uniformity explicit. If the current draft leaves any ambiguity on this point, the revision will remove it. revision: yes

Circularity Check

0 steps flagged

No circularity; phase transition derived from explicit Wiener-like measures on C[0,1]

full rationale

The paper defines h_φ(λ) directly as the integral against the Bernoulli convolution μ_λ and states sufficient conditions plus a phase-transition result for almost-every φ drawn from specified Wiener-like measures. These measures are introduced as an external construction on C[0,1]; the 'almost every' quantifier is therefore an explicit hypothesis rather than a fitted parameter or self-referential definition. No equations reduce a claimed prediction to a prior fit, no uniqueness theorem is imported from self-citation, and the Hölder restriction is stated up-front without being smuggled in. The derivation chain therefore remains self-contained against external analytic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard background facts about Bernoulli convolutions and function spaces; no free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption Standard definition and basic properties of Bernoulli convolution measures μ_λ
The abstract assumes the reader knows the construction of μ_λ for λ in (0,1).
domain assumption Hölder continuity of the observables φ
The regularity statements are stated for Hölder φ.

pith-pipeline@v0.9.0 · 5408 in / 1261 out tokens · 70442 ms · 2026-05-10T09:22:11.074290+00:00 · methodology

discussion (0)

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