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Fourier decay and non-decay for pseudo-affine self-conformal measures

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Pith's one-line read The paper constructs a C∞ self-conformal iterated function system whose derivative is constant on its attractor but which is not C^1-conjugate to any self-similar system, and shows its uniform stationary measure is not Rajchman; separately,

desk verdict Careful sharpness results at the AHW23 boundary; the Fourier analysis is solid, but the construction's foundations live in an unpublished companion paper. read the letter →

arxiv 2607.22001 v1 pith:XVP47DFE submitted 2026-07-24 math.DS math.CA

Amir Algom , Snir Ben Ovadia , Federico Rodriguez Hertz , Mario Shannon This is my paper · ORCID
classification math.DSmath.CA MSC 28A8037C4542A38
keywords pseudo-affineIFSself-conformalmeasuresRajchmanpropertyFourierdecaydynamicalproportionsBernoulliconvolutionsPisotnumbersmatrixRieszproducts
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 how sharp recent sufficient conditions for polynomial Fourier decay of self-conformal measures actually are. It proves two things. First, there exists a C∞ iterated function system that is linear in the sense that all derivatives on the attractor equal one constant, yet is not C^1-conjugate to any self-similar system, and its uniform (1/2,1/2) stationary measure has a Fourier transform that does not vanish at infinity. This shows that a known C^2 Fourier-decay criterion, which requires failure of conjugacy to a linear system, cannot be relaxed to failure of conjugacy to a self-similar system at this regularity. Second, for every λ in (0,1/2), it constructs random C^{r,α} diffeomorphisms whose derivatives are constant on the support of the Bernoulli convolution μ_λ, but whose push-forwards μ_λ nonetheless enjoy polynomial Fourier decay, even when μ_λ itself is not Rajchman. The proofs rely on pseudo-affine IFS and replace the classical infinite-convolution product with a matrix-valued Riesz product.

What carries the argument

The central object is the pseudo-affine IFS: an IFS in which every map's derivative equals the same λ at every point of the attractor. Its geometry is encoded in the dynamical proportions, the ratios between a gap and its two descendant gaps; regularity of the system is characterized by how fast these proportions converge to λ, and conjugacy between two pseudo-affine systems is characterized by the pointwise convergence of the ratio of their cocycles. For the Fourier arguments, the central identities are the matrix Riesz product for the two-state renewal recursion (used in Theorem 1.1) and the level-homogeneous independent cosine product for the random construction (used in Theorem 1.2). The

What would settle it

Compute the cylinder-length ratios of the constructed IFS from Proposition 2 at all finite words: the ratio Ψ(w)/λ^{|w|} must equal 1 except at the root word, where it equals κ/λ. If the ratios deviate at any non-root word, the cocycle-criterion argument for non-conjugacy is void. Separately, numerically evaluate the Fourier transform of the stationary measure at the frequencies λ^{-k} for large k; the proof predicts a fixed positive lower bound, so observing decay to zero would indicate an error in the matrix-product estimate.

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Extended reading notes

Core claim

The central claim is that two phenomena coexist. On the negative side, a C∞ pseudo-affine IFS with slope λ exists whose dynamical proportions equal λ except at the root word, where a single defect κ=λ+ε is inserted. The defect vanishes at small scales, yielding C∞ smoothness, but it persists in the dynamical proportions, so the cocycle-ratio criterion from the companion theory shows the IFS is not C^1-conjugate to the self-similar system Φ_λ. The uniform stationary measure is not an infinite convolution; its finite approximations satisfy a two-state renewal recursion, and passing to Fourier transforms gives a matrix product whose factors are 2×2 matrices with phases concentrated near integer

Load-bearing premise

The entire construction relies on three theorems from the authors' companion paper: that prescribed dynamical proportions are realizable by a C^{r,α} pseudo-affine IFS with matching regularity, and that the convergence of the cocycle-ratio function is both necessary and sufficient for conjugacy to exist; if any of these has a flaw, both Theorem 1.1 and the conjugacy and self-conformal-structure parts of Theorem 1.2 collapse.

Editorial extensions

If this is right

  • The known C^2 criterion for polynomial Fourier decay of self-conformal measures is sharp in its regularity setting: at C^1 conjugacy level, absence of conjugacy to a self-similar system does not imply the Rajchman property.
  • Pseudo-affine IFS provide an exactly solvable class where regularity, conjugacy, and Fourier behaviour can be read off from a single sequence of gap-length ratios.
  • For Bernoulli convolutions with contraction ratio below 1/2, the standard phenomenon that smooth nonlinear images have polynomial Fourier decay persists even when the image map's derivative is flat on the support, so L^2-flattening is not necessary in that regime.
  • If λ^{-1} is Pisot, no C∞ diffeomorphism whose derivative is constant on the support of μ_λ can yield polynomial Fourier decay; the best possible in this class is stretched-exponential decay, which the paper realizes.
  • The construction yields an explicit, computable lower bound on the Fourier-decay exponent in terms of λ and the smoothness exponent s.

Reading between the lines

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

  • If the companion realizability theorems are sound, a deterministic version of the random construction should be obtainable by replacing the Borel–Cantelli step with an equidistribution or Diophantine approximation argument on the perturbation variables, since the level-homogeneous structure makes the induced geometry particularly rigid.
  • The matrix Riesz product for two-state recursions likely extends to IFS with more than two maps, yielding block-matrix products whose decay is governed by the joint spectral radius of the associated matrices; the Pisot-type argument here suggests a general arithmetic obstruction to the Rajchman property when all phases are exponentially close to integers.
  • Proposition 9 places a ceiling on the decay rate for C∞ flat derivatives in the Pisot case; whether the polynomial-to-stretched transition at C∞ is sharp in the non-Pisot case is a natural next question.
  • These examples indicate that 'lack of conjugacy to self-similar' is not the right invariant for Fourier decay at finite regularity; the more relevant quantity may be the rate at which the dynamical proportions converge to their limiting value, rather than whether that limit system is affine.
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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 / 4 minor

Summary. The paper studies sharpness of sufficient conditions for polynomial Fourier decay of self-conformal measures on the line. Theorem 1.1 constructs a C^∞ pseudo-affine IFS Φ on [0,1] that is not C^1-conjugate to a self-similar IFS, yet its uniform Cantor–Lebesgue measure is not Rajchman; this shows that the 'linear' hypothesis in [AHW23] cannot be replaced by 'self-similar' at C^1 regularity. Theorem 1.2 shows that for every λ∈(0,1/2), r∈N, and α∈(0,1], there exists a C^{r+α} diffeomorphism g with g' constant on supp μ_λ such that g μ_λ has polynomial Fourier decay with an explicit exponent, even when μ_λ is non-Rajchman because λ^{-1} is Pisot. The paper also proves a matching Pisot obstruction (Prop. 9), a C^∞ stretched-exponential variant (Prop. 10), and a non-affine conjugacy preserving non-decay (Prop. 11). The proofs use a matrix Riesz-product representation for the Fourier transform, a one-scale cosine moment estimate, and a Borel–Cantelli net argument, all built on the pseudo-affine framework of [ABRS26].

Significance. If the companion results are valid, Theorem 1.1 is a substantial contribution: it provides the first example separating C^1-conjugacy to self-similar from the linearity criterion, and Proposition 9 gives a sharp regularity threshold for the phenomenon in Theorem 1.2. The Fourier-analytic arguments in Sections 3.3 and 4.2 are coherent and largely self-contained; the matrix Riesz product and the one-scale cosine moment bound are elegant, and the constants in Lemma 1 and Proposition 8 are carefully tracked. The main vulnerability is the paper's heavy dependence on the to-appear companion [ABRS26]: all of the dynamical constructions and the conjugacy conclusions rest on Theorems 2.2 and 2.3 stated there. The paper also frankly acknowledges the overlap with Ekström [Eks16] for the existence part of Theorem 1.2, while adding an explicit self-conformal structure and a C^∞ variant.

major comments (2)
  1. [Section 2.2 (Theorems 2.1–2.3)] The central results are conditional on theorems that are stated but not proved. Theorem 2.2 is used to construct the IFS Φ in Proposition 2 (Theorem 1.1) and the random IFS Φ_ω in Proposition 7 (Theorem 1.2); Theorem 2.3 is used to conclude non-conjugacy in Theorem 1.1 and to derive the derivative identity h'_ω ≡ 1/L_0(ω) in Proposition 7. These are load-bearing: if any of these statements is false or has a hidden hypothesis, the main theorems fail. Since [ABRS26] is to appear, the submitted manuscript is not independently verifiable. The authors should either include full proofs of the needed parts of Theorems 2.1–2.3, or make the companion manuscript available to referees, or state precisely which parts of the present results are unconditional without it.
  2. [Section 3.1 (proof of Proposition 2)] The reduction from an arbitrary C^1 conjugacy to the canonical address-preserving map is only implicit. In the proof of Proposition 2, after normalizing by an affine map, the authors apply Theorem 2.3 to show that Φ is not C^1-conjugate to Φ_λ. This is valid because any conjugacy on the attractor sends cylinders to cylinders and therefore agrees with the canonical map on the attractor; however, this fact is not stated explicitly in the proof. Since the 'if and only if' in Theorem 2.3 is for the canonical map, the authors should include a sentence explaining that every conjugacy on the attractor is the canonical map (up to the affine normalization), so the criterion applies. This is a presentation point, but it is load-bearing for the non-conjugacy conclusion.
minor comments (4)
  1. [Section 2.1] Typo: '0=f_0(1)<f_0(1)<...' should presumably be '0=f_0(0)<f_0(1)<...'.
  2. [Equation (26)] The notation M_j for negative j uses λ^j = q^j with j<0; this is implicit and should be stated explicitly, since Lemma 1 relies on it.
  3. [Section 1.2 / Remark 1] The overlap with Ekström [Eks16] is handled honestly, but the abstract and introduction state Theorem 1.2 as if the existence part were new. Consider moving the acknowledgement of the overlap to the introduction or making the incremental nature explicit in the theorem statement.
  4. [Proposition 9] The induction argument for h^{(j)}(0)=0 when r≥2 is compressed. It would be clearer to spell out that h' is constant on a perfect set, so the derivative of h' at each point of the set vanishes, and then iterate.

Circularity Check

1 steps flagged · score 4.0 of 10

Main geometric construction and non-conjugacy are imported from the authors' companion paper [ABRS26]; the Fourier arguments are internally coherent.

  1. self citation load bearing [Section 1.3; Section 2.2 (Theorems 2.1–2.3); applied in Prop. 2 (Section 3.1) and Prop. 7 (Section 4.1)]
    "In [ABRS26] we solved this problem, by introducing pseudo-affine IFS ... and showing that there exist such IFS that are not C^1-conjugated to self-similar, for every s ∈ [1,∞]. ... We summarize the main properties of pseudo-affine IFS that will be used in this paper in Section 2.2, this time rigorously and with full details. All of these were developed in [ABRS26]."

    Theorems 2.1–2.3 are quoted from [ABRS26] (Propositions 12,13,15; Proposition 15; Theorem 20), all by the same four authors and 'to appear'. Proposition 2 obtains the C∞ pseudo-affine IFS and its non-conjugacy by invoking Theorem 2.2 ('Theorem 2.2 therefore yields...') and Theorem 2.3 ('By Theorem 2.3, Φ is not C^1-conjugate to Φ_λ'). Proposition 7 similarly obtains the C^{r,α} IFS, the conjugacy h_ω, and h'_ω ≡ 1/L_0 by citing the same theorems. Thus the load-bearing existence/regularity/conjugacy content of Theorems 1.1 and 1.2 is not derived in this paper; it is a self-citation. If any of those companion results fails, the main geometric claims fail. This is load-bearing self-citation rather than a definitional or fitted-input circularity.

full rationale

The Fourier-decay parts of the paper are not circular. In Theorem 1.1 the non-Rajchman conclusion is obtained from a genuine matrix Riesz-product limit: Proposition 6 and Eq. (32) show bμ(λ^{-k}) → ρ with |ρ−1| ≤ e^S−1 < 1/2, so ρ ≠ 0. This is an internal arithmetic argument. In Theorem 1.2 the polynomial decay is proved by the random construction's independent cosine factors plus the one-scale moment estimate (Lemma 3), Markov's inequality, and Borel–Cantelli; the exponent is a derived quantity, not fitted to the tested measure. The paper also explicitly disclaims novelty for the bare existence part of Theorem 1.2 with respect to Ekström. The genuine circularity concern is structural: the pseudo-affine framework, the realizability theorem (2.2), and the conjugacy criterion (2.3) are all imported from the authors' own companion paper [ABRS26], to appear, and the central non-conjugacy claim in Theorem 1.1 and the C^{r,α} conjugacy in Theorem 1.2 reduce to those citations. No proof of these theorems is included here and no independent check is supplied. This warrants a moderate score of 4: the central Fourier content is independent, but the geometric/conjugacy premise is load-bearing self-citation.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

No new postulates: the pseudo-affine IFS framework is from the authors' prior work [ABRS26], and the random Cantor sets are explicitly constructed from given random variables. The free parameters are visible construction parameters, not hidden data fits.

free parameters (6)
  • q (integer, chosen large) = λ = 1/q; in Theorem 1.1, λ^{-1} must be a Pisot number and q large enough so e^S−1 < 1/2 (S ≈ 12πλ).
    Determines the slope of the pseudo-affine IFS. The choice q large ensures the perturbation sums are small in Lemma 2.
  • N (integer ≥2) = ε = λ^N/(1−2λ−λ^N) in Theorem 1.1.
    Defect depth in the dynamical proportions; chosen so β = λ+λ^N and A,B become integer polynomials in λ, required for the Pisot arithmetic.
  • c (perturbation amplitude) = c ∈ (0,1/2) with a_n = c λ^{(s−1)n} in Theorem 1.2; chosen small enough to keep q_n < 1/2 uniformly.
    Controls the size of random perturbations of the homogeneous Cantor set construction.
  • s = r+α (regularity order) = User-specified: 1≤r<∞, 0<α≤1.
    The theorem is stated for every finite smoothness; the exponent τ(s,λ) depends on s.
  • δ, c_0, ϑ (Proposition 10 parameters) = a_n = c_0 λ^{δ n log(n+2)}, with 0<ϑ<1 and δ < (1−ϑ)/(16 log β).
    Superexponential perturbation parameters that yield the C∞ stretched-exponential variant.
  • Finitely supported sequence (a_n) in Z[q^{-1}] (Proposition 11) = Non-zero, small sup-norm.
    Used to build a non-affine C∞ diffeomorphism whose pushforward and original measure are both non-Rajchman.
assumptions (4)
  • domain assumption Theorems 2.1–2.3 of [ABRS26]: prescribed dynamical proportions satisfying the stated decay rates are realizable by a C^{r,α} pseudo-affine IFS; regularity is characterized by the decay rate; C^s conjugacy between pseudo-affine IFSs is equivalent to uniform convergence of the cocycle ratio (10).
    Load-bearing for both main theorems. These results are from the authors' own to-appear paper and are not proved in the submission.
  • standard math Standard coding/transfer facts for hyperbolic, strongly separated IFSs: existence and uniqueness of the attractor, the coding map, and the stationary measure; Proposition 1 (gap-length formula) from [ABRS26, Lemma 8].
    Used throughout Sections 2–4 to reduce geometric quantities to dynamical proportions.
  • standard math Erdős–Salem theory: for λ^{-1} a Pisot number, the product ∏_{n≥1} cos(πλ^n) converges to a positive number and ∥λ^{-k}∥ decays exponentially; the Bernoulli convolution μλ is not Rajchman and satisfies the lower bound (67).
    Invoked in the non-decay proof (Thm 1.1) and in the Pisot obstruction (Prop 9).
  • domain assumption For 0<λ<1/2, the IFS Φλ = {λx, λx+1−λ} is strongly separated and the Bernoulli convolution μλ is the unique (1/2,1/2)-stationary measure.
    The strong separation guarantees the product/Riesz representation for the Fourier transform used in Section 4.

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Pith. "Pith review of Fourier decay and non-decay for pseudo-affine self-conformal measures." pith.science (2026). https://pith.science/paper/XVP47DFE

@misc{pith2026260722001,
  author       = {Pith},
  title        = {Pith review of: Fourier decay and non-decay for pseudo-affine self-conformal measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XVP47DFE}},
  note         = {Machine review of arXiv:2607.22001}
}
abstract

We study the sharpness of recent sufficient conditions for polynomial Fourier decay of self conformal measures on the line. First, we construct a $C^\infty$ iterated function system which is not $C^1$-conjugate to self-similar, but which nevertheless admits a stationary measure that is not Rajchman. Second, for every strongly separated Bernoulli convolution $\mu$ and every $1\leq r<\infty$, we construct a $C^r$-diffeomorphism $h$ such that $h'$ is constant on $\operatorname{supp}\mu$, yet the image measure $h\mu$ has polynomial Fourier decay. All constructions are within the framework of pseudo-affine iterated function systems, previously introduced by the authors.

Figures

Figures reproduced from arXiv: 2607.22001 by the authors.

Figure 1
Figure 1. The increments in the random orbit (xn)n≥0. Consequently, setting A := λ(1−β), B := (1−λ)β, we have |Xw | −|Xw1| =    λ n−1A, wn = 0, λ n−1B, wn = 1. Proof. For a nonempty word w, the definition of the dynamical proportions in (11) gives Ψ(w) =    λ |w| , if the last symbol of w is 0, κλ|w|−1 , if the last symbol of w is 1. Therefore, P w∈W Ψ(w) = 1 + P∞ n=1 2 n−1 ¡ λ n + κλn−1 ¢ = 1 + λ+κ 1−2λ . Proposition 1… view at source ↗

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