Smoothness of random self-similar measures on the line and the existence of interior points

Bal\'azs B\'ar\'any; Micha{\l} Rams

REVIEW 2 major objections 4 minor 1 cited by

Smoothness of random self-similar measures on the line and the existence of interior points

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that, under a Fourier-decay condition on a random perturbation, any symbolic measure with local dimension greater than 1 pushes forward to an absolutely continuous measure with Hölder-continuous density almost surely…

desk verdict A real new Hölder-density theorem for random self-similar measures, but the central estimate in Section 3 has a gap that needs fixing before the proof is accepted. read the letter →

arxiv 2412.06008 v2 pith:PY6ORCG2 submitted 2024-12-08 math.DS math.PR

Balázs Bárány , Micha{l} Rams This is my paper · ORCID

classification math.DSmath.PR MSC 28A8060G3060G57

keywords randomself-similarsetsiteratedfunctionsystemsabsolutecontinuityHöldercontinuousdensityinteriorpointFouriertransformlocaldimensionperturbation

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

Self-similar iterated function systems on the line have a similarity dimension $s$; when $s>1$, the paper asks whether a random perturbation of the system smooths the projected measure. The answer is yes under two conditions: the symbolic measure must satisfy the cylinder bound $\mu([\omega])\le K\lambda_\omega^{s'}$ with $s'>1$, and the perturbation's distribution must have Fourier decay $|\hat\Theta(x)|\le C(1+|x|)^{-M}$ with $M\ge s'$. Then $\nu^\Theta=(\Pi^\Theta)_*\mu$ is almost surely absolutely continuous with respect to Lebesgue measure and has a Hölder-continuous density. Because a continuous density that is positive somewhere forces an open interval in the support, the same theorem gives an interior point of the random self-similar set whenever $s>1$ and $M\ge s$. This extends earlier interior-point results that were confined to uniform perturbations and yields regularity of the density, not only its existence.

What carries the argument

The argument runs through the random perturbation applied independently at every finite word of the symbolic tree, $\Pi^\Theta(\omega)=\sum_{j\ge0}\lambda_{\omega_j}(t_{\omega_j}+\Theta_{\omega_j})$, and through the Fourier-domain quantity $W(\xi_1,\dots,\xi_p)=|\int\prod_{k=1}^p\widehat{\nu^\Theta}(\xi_k)\,d\mathbb{P}(\Theta)|$. Because the perturbations at different nodes are independent, this quantity factorizes over the tree, and the Fourier decay assumption bounds each factor by $(1+|\lambda\xi|)^{-M}$. A combinatorial estimate over the first levels at which the $p$ sampled symbolic sequences separate shows that $\int\prod_{k=1}^p|\xi_k|^\gamma W(\xi_1,\dots,\xi_p)\,d\xi_1\cdots d\xi_p$ is finite whenever $\gamma<s'-1$ and $p$ is chosen large and even. That integrability is what converts the Fourier-decay condition into the polynomial moment estimate $\mathbb{E}|\vartheta(a)-\vartheta(b)|^p\le C|a-b|^{p\gamma}$, the input for the Hölder-continuity criterion that delivers the continuous density.

What would settle it

A concrete check is to take the IFS with maps $f_1(x)=x/2$, $f_2(x)=x/2+1/3$, $f_3(x)=x/2+2/3$ (similarity dimension $\log 3/\log 2>1$), the natural Bernoulli measure, and a compactly supported smooth perturbation with fast Fourier decay, then compute the random density by Fourier inversion at many nearby pairs of points over many realizations. The theorem predicts a Hölder-continuous density almost surely; any realization whose computed density has unbounded oscillations or fails to satisfy a power-law modulus at fine scales would contradict the central claim.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for any self-similar IFS with similarity dimension $s>1$, any measure $\mu$ on the symbolic space with $\mu([\omega])\le K\lambda_\omega^{s'}$ for some $s'>1$, and any perturbation $\Theta$ satisfying $|\hat\Theta(x)|\le C(1+|x|)^{-M}$ with $M\ge s'$, the projected measure $\nu^\Theta=(\Pi^\Theta)_*\mu$ is absolutely continuous and its density is Hölder continuous almost surely. The proof first obtains almost-sure absolute continuity with an $L^2$ density, then uses one-dimensional Fourier inversion to represent the density pointwise as a limit of truncated inverse Fourier integrals, and finally views the density as a stochastic process in the spatial variable $x$. A moment bound of the form $\mathbb{E}|\vartheta(a)-\vartheta(b)|^p\le C|a-b|^{p\gamma}$ with $p\gamma>1$ feeds a Hölder-continuity criterion from stochastic process theory, which produces a continuous version of the density. Corollary 1.2 applies the theorem to the natural Bernoulli measure with weights $\lambda_i^s$; since this measure satisfies the assumption with $s'=s$, the random attractor $\Lambda^\Theta$ contains an interior point almost surely.

Load-bearing premise

The proof rests on the Fourier decay assumption $|\hat\Theta(x)|\le C(1+|x|)^{-M}$ with $M\ge s'$; the most natural perturbation, uniform on an interval, does not satisfy it, so the theorem leaves that case open.

Editorial extensions

If this is right

Corollary 1.2: whenever the similarity dimension $s>1$ and the Fourier decay exponent satisfies $M\ge s$, the random self-similar set $\Lambda^\Theta$ contains an open interval almost surely.
The density of $\nu^\Theta$ is not merely absolutely continuous but Hölder continuous, so it has a continuous representative defined at every point of the line.
The result applies to every symbolic measure satisfying the cylinder bound (1.4), not only to the natural Bernoulli measure, as long as its symbolic local dimension is strictly above 1.
If the symbolic local dimension drops to 1 or below, no bounded density can exist; the paper shows the condition $s'>1$ is close to optimal for a bounded density, not merely sufficient.
The proof also confirms, through the Fourier representation, that the $L^2$-based absolute continuity of the projected measure is upgraded to a pointwise continuous object almost surely.

Reading between the lines

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

One natural extension is to test whether the same Hölder conclusion holds for the uniform perturbation that the theorem excludes; a numerical experiment with a smooth approximation to the uniform distribution would show whether the Fourier decay condition is truly necessary or merely a technical convenience.
The one-dimensional Fourier-inversion step appears to be essential; in higher dimensions the analogous almost-everywhere inversion is false, so extending the result to random self-affine sets on $\mathbb{R}^d$ would likely require a different mechanism and may fail.
Since the theorem works for any measure satisfying (1.4), the interior-point corollary should also hold for any symbolic measure with full support and local dimension above 1, not just the natural Bernoulli measure.
The proof's dependence on $\gamma<s'-1$ suggests the Hölder exponent one can extract is controlled by the excess of the symbolic dimension over 1, so measures with larger $s'$ should yield smoother densities; this quantitative dependence is not stated as a theorem and would be worth checking.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the paper has a real new theorem—Hölder continuous density for projected random self-similar measures under the Fourier decay condition (1.3)—but the main estimate in Section 3 is not justified as written. The stress-test note is right, and it lands on a load-bearing step.

What is genuinely good: Theorem 1.1 is new and not in the cited papers. Carleson plus Kolmogorov is a sensible route, the measurability lemma is handled carefully, and the authors are honest that (1.3) excludes the uniform perturbation studied in [5]. No circularity, no data fitting. Corollary 1.2 is a modest repackaging of the interior-point result from [5], but as a corollary it is fine.

Where the proof breaks: the change-of-variables step around (3.5)–(3.6) assumes |ξ_k| can be treated like |w_k| after passing to w_k = Σ_{ℓ∈N_k} ξ_ℓ. The triangular matrix has an off-diagonal inverse. For a realized nested splitting with p=4, N1={1,2,3,4}, N2={2,3,4}, N3={3,4}, N4={4}, the slice w1=w2=w3=0, w4=t gives ξ4=t, ξ3=-t, and the integrand decays like |t|^{2γ}(1+λ4|t|)^{-M}. That integral needs M>1+2γ, not M>1+γ. The text explicitly chooses γ

This is repairable: choose γ < (s′−1)/2 and then p large enough that pγ>1. The counting estimate (3.7) would need to be rechecked under that condition, but it looks like it can absorb it. As written, the proof overclaims its constant regime.

Bottom line: this deserves a serious referee, but not acceptance yet. The main idea is likely right, the gap is localized, and the authors should be asked to fix Section 3 and state the M,γ condition precisely.

Referee Report

2 major / 4 minor

Summary. The paper studies randomly perturbed self-similar iterated function systems on the line. The main result (Theorem 1.1) asserts that if the similarity dimension satisfies s > 1 and a symbolic measure satisfies the cylinder bound μ([ω]) ≤ K λ_ω^{s'} with s' > 1, then for a random perturbation whose Fourier transform decays like (1+|x|)^{-M} with M ≥ s', the projected measure is almost surely absolutely continuous with Hölder continuous density. Corollary 1.2 concludes that the randomly perturbed attractor contains an interior point almost surely when M ≥ s. The proof combines Carleson's theorem with Kolmogorov's continuity theorem: it estimates p-th moments of the density process, using a change of variables and a counting argument over the levels at which the p symbolic sequences first separate. The paper also contains a measurability lemma and relies on earlier results of Jordan–Pollicott–Simon and Gu–Miao for the almost sure absolute continuity with L² density.

Significance. If the main theorem is correct, it would extend recent work of Dekking–Simon–Székely–Szekeres and Gu–Miao by producing Hölder regularity of the density rather than only absolute continuity or dimension information, and it would give a new proof of the interior-point corollary. The method is interesting: the use of Carleson's theorem restricts the argument to dimension one, and the authors state this limitation explicitly. The proof is not circular and relies on established external results. The authors also explicitly acknowledge that the most natural uniform-on-an-interval perturbation does not satisfy the Fourier decay condition (1.3), which is an honest limitation. However, the central change-of-variables estimate in Section 3 is not justified as written and needs correction; with a local repair the theorem may stand, but the present proof is incomplete.

major comments (2)

[§3, estimate preceding (3.6)] The displayed estimate that leads to (3.6) is false under the stated assumption M = s' > 1 + γ. After the change of variables w_k = λ_{ω(k),N_k} Σ_{ℓ∈N(ω(k),N_k)} ξ_ℓ, one has ξ = U^{-1} D^{-1} w, so the factor ∏ |ξ_k|^γ introduces powers λ_{ω(k),N_k}^{-γ} that are not accounted for in the text. For a configuration with p = 2, N_1 = 0, λ_{ω(1),N_1} = 1, and λ_{ω(2),N_2} = λ, the line w_1 = 0, w_2 = t gives ξ_1 = -t/λ and ξ_2 = t/λ. The transverse integral over w_1 contributes λ^{-1} from the Jacobian and a bounded constant, leaving an integral of order λ^{-1-2γ} ∫ |t|^{2γ} (1+|t|)^{-M} dt. This converges only if M > 1 + 2γ, not merely M > 1 + γ. Since the text later chooses γ < s' - 1, there are admissible parameters (e.g., s' = 1.4, γ = 0.3, M = 1.4) for which the asserted bound fails. The estimate must be reworked: using the correct exponent 1 + 2γ (or else the full product W rather than the crude bound by the selected N_k factors) and then choosing γ < (s' - 1)/2 makes the counting estimate (3.7) converge, but this is not what the manuscript proves.
[§3, final choice of constants] The final choice of constants ('choose γ < s' - 1, then choose some even p large enough that pγ > 1') is tied to the erroneous exponent. With the corrected estimate, one must require M > 1 + 2γ, and since the theorem only assumes M ≥ s', the admissible choice is γ < (s' - 1)/2. This is compatible with the later requirement pγ > 1 by taking p large, but the current text does not state this condition. In addition, the sentence 'we also assume M = s' > 1 + γ' should be replaced by a precise statement such as 'M ≥ s' > 1 + 2γ' after fixing γ, and the summation in (3.7) should be reread with the exponent s' - 1 - 2γ > 0.

minor comments (4)

[§3, equation (3.7)] Equation (3.7) has a notational mismatch: the left-hand side is written for ∏_{k=1}^p λ_{ω(k),N_k}, while the right-hand side uses ℓ and K^{ℓ-1}; please define ℓ = p and use the same symbol throughout.
[§3, notation] The sets N_{ω(1),...,ω(p)}(τ^n) and N(ω(k),N_k) are used before being defined explicitly; a short formal definition would prevent confusion, especially because the change-of-variables argument depends on their exact meaning.
[Throughout] There are numerous typographical issues (e.g., 't he' in the title, 'Micha/suppress l' in the author line, 'e very' and 'con tains' in the abstract, 'te function' in the proof of Theorem 1.1, and a missing closing parenthesis in the proof of Lemma 2.4). These should be corrected in a revised version.
[Theorem 1.1 and §3] The theorem assumes M ≥ s', but the proof says 'M = s''. Since larger M gives stronger decay, one can take the worst case, but the text should say so explicitly to avoid the impression that the theorem's hypothesis is changed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof derives Hölder continuity from the stated Fourier-decay and symbolic-measure assumptions using external theorems.

full rationale

The paper's derivation chain is self-contained with respect to circularity. Theorem 1.1 assumes the symbolic measure condition (1.4) and the Fourier decay condition (1.3), and then proves absolute continuity with Hölder density via Carleson's theorem, Kolmogorov's continuity theorem, and an estimate on the p-fold integral of the Fourier transform. The density limit in (3.1) and Carleson inversion in (3.2) are standard external facts, not restatements of the desired conclusion. Proposition 2.3, which supplies L2 density, is cited from Jordan, Pollicott and Simon [12, Proposition 4.4(b)]; that is an independent prior result and is not used to force the Hölder regularity. The constants γ and p are chosen after the estimates to satisfy 1+γ < s' and pγ > 1, so they are not fitted to the target result. The citation of Dekking, Simon, Székely and Szekeres [5] for the prior interior-point theorem is contextual and not load-bearing: Corollary 1.2 is proved from Theorem 1.1, not from [5]. The skeptic's concern about the change-of-variables estimate in Section 3 is a possible technical gap in the proof of (3.5)-(3.6), not a circularity: it concerns whether the integrand decays as claimed under the stated choice of M, not whether an input is assumed equal to the output. No self-definitional step, fitted-input prediction, or renaming of a known result occurs in the argument.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. The random perturbation Θ is a standard device from prior literature (e.g., Jordan, Pollicott and Simon). Free parameters γ and p are proof-technical choices, not fitted values. All substantive assumptions are stated explicitly as hypotheses (1.3) and (1.4), and background theorems are cited.

free parameters (2)

γ (Hölder exponent parameter) = any number in (0, s'-1)
Chosen in Section 3 to make the Kolmogorov exponent pγ>1 and the summability condition Σ (max λ_j)^{m(s'-1-γ)} finite. It is an ad hoc existence choice, not fitted to data.
p (moment order) = even integer with pγ>1
Chosen in Section 3 to apply Kolmogorov's continuity theorem with exponent α=pγ>1. Not fitted to data; it is a proof parameter.

assumptions (6)

standard math Carleson's theorem: L2 functions on R are recovered from their Fourier transform at Lebesgue-almost every point.
Invoked as Theorem 2.1 and used in (3.2) to represent the density ϑ as a limit of inverse Fourier integrals.
standard math Kolmogorov's continuity theorem: a process with E|X_a-X_b|^p ≤ C|a-b|^α, α>1, has a Hölder continuous modification.
Invoked as Theorem 2.2 and used in Proposition 3.3 to obtain a continuous version of the density field.
domain assumption Proposition 2.3 (from Jordan, Pollicott and Simon [12, Prop 4.4(b)]): under condition (1.4), ν^Θ is almost surely absolutely continuous with L2 density.
External result used to ensure the density exists and to justify the Fourier representation and Carleson's theorem.
domain assumption Fourier decay condition (1.3): |\hatΘ(x)| ≤ C(1+|x|)^{-M} with M ≥ s'.
Hypothesis of Theorem 1.1; used in the key integral estimate in Section 3 to control the decay of W.
domain assumption Symbolic local dimension condition (1.4): μ([ω]) ≤ K λ_ω^{s'} for all finite words, with s'>1.
Hypothesis of Theorem 1.1; used in the counting estimates for Wℓ in Section 3.
standard math Fubini, Fatou, Luzin, and Egorov theorems.
Used in Lemmas 2.4, 3.2 and Proposition 3.3 to justify measurability, exchange of integrals, and pointwise limits.

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Pith. "Pith review of Smoothness of random self-similar measures on the line and the existence of interior points." pith.science (2026). https://pith.science/paper/PY6ORCG2

@misc{pith2026241206008,
  author       = {Pith},
  title        = {Pith review of: Smoothness of random self-similar measures on the line and the existence of interior points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PY6ORCG2}},
  note         = {Machine review of arXiv:2412.06008}
}

read the original abstract

In this paper, we study the smoothness of the density function of absolutely continuous measures supported on random self-similar sets on the line. We show that the natural projection of a measure with symbolic local dimension greater than 1 at every point is absolutely continuous with H\"older continuous density almost surely. In particular, if the similarity dimension is greater than 1 then the random self-similar set on the line contains an interior point almost surely.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On the Fourier transform of random Bernoulli convolutions
math.DS 2025-07 accept novelty 7.0 of 10

For random Bernoulli convolutions, the Fourier transform is in L^1 almost surely whenever λ_g > 2/π, giving absolute continuity and non-empty interior; polynomial Fourier decay holds for every λ_g.

Reference graph

Works this paper leans on

20 extracted references · 18 canonical work pages · cited by 1 Pith paper

[5]

Dekking, K

M. Dekking, K. Simon, B. Sz´ ekely, and N. Szekeres. The interior of randomly perturbed self-similar sets on the line. Adv. Math., 448:Paper No. 109724, 43, 2024

work page 2024
[1]

B´ ar´ any, K

B. B´ ar´ any, K. Simon, and B. Solomyak. Self-similar and self-aﬃne sets and measures , volume 276 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, [2023] ©2023

work page 2023
[2]

V. I. Bogachev. Measure theory. Vol. I, II . Springer-Verlag, Berlin, 2007

2007
[3]

Cawley and R

R. Cawley and R. D. Mauldin. Multifractal decompositions of Moran fractals. Adv. Math., 92(2):196– 236, 1992

work page 1992
[4]

Cs¨ ornyei, T

M. Cs¨ ornyei, T. Jordan, M. Pollicott, D. Preiss, and B. Solomyak . Positive-measure self-similar sets without interior. Ergodic Theory Dynam. Systems , 26(3):755–758, 2006

work page 2006
[6]

Erraoui and Y

M. Erraoui and Y. Hakiki. Images of fractional Brownian motion w ith deterministic drift: positive Lebesgue measure and non-empty interior. Math. Proc. Cambridge Philos. Soc. , 173(3):693–713, 2022

work page 2022
[7]

J. M. Fraser and P. Shmerkin. On the dimensions of a family of over lapping self-aﬃne carpets. Ergodic Theory Dynam. Systems , 36(8):2463–2481, 2016

work page 2016
[8]

Grafakos

L. Grafakos. Modern Fourier analysis , volume 250 of Graduate Texts in Mathematics . Springer, New York, second edition, 2009. 10

work page 2009

Show all 20 references

[9]

Gu and J

Y. Gu and J. J. Miao. Generalized q-dimensions of measures on nonautonomous fractals. Preprint, available at arXiv:2411.17298 , 2024

2024 arXiv
[10]

M. Hochman. On self-similar sets with overlaps and inverse theor ems for entropy. Ann. of Math. (2) , 180(2):773–822, 2014

2014
[11]

J. E. Hutchinson. Fractals and self-similarity. Indiana Univ. Math. J. , 30(5):713–747, 1981

1981
[12]

Jordan, M

T. Jordan, M. Pollicott, and K. Simon. Hausdorﬀ dimension for ra ndomly perturbed self aﬃne attractors. Comm. Math. Phys. , 270(2):519–544, 2007

2007
[13]

Jordan and A

T. Jordan and A. Rapaport. Dimension of ergodic measures pro jected onto self-similar sets with overlaps. Proc. Lond. Math. Soc. (3) , 122(2):191–206, 2021

2021
[14]

Keane, M

M. Keane, M. Smorodinsky, and B. Solomyak. On the morphology of γ-expansions with deleted digits. Trans. Amer. Math. Soc. , 347(3):955–966, 1995

1995
[15]

Khoshnevisan

D. Khoshnevisan. Multiparameter processes. Springer Monographs in Mathematics. Springer-Verlag, New York, 2002. An introduction to random ﬁelds

2002
[16]

Koivusalo

H. Koivusalo. Dimension of uniformly random self-similar fractals. Real Anal. Exchange, 39(1):73–90, 2013/14

2013
[17]

P. Mattila. Geometry of sets and measures in Euclidean spaces , volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rect iﬁability

1995
[18]

Peres, K

Y. Peres, K. Simon, and B. Solomyak. Absolute continuity for ra ndom iterated function systems with overlaps. J. London Math. Soc. (2) , 74(3):739–756, 2006

2006
[19]

Saglietti, P

S. Saglietti, P. Shmerkin, and B. Solomyak. Absolute continuity o f non-homogeneous self-similar measures. Adv. Math., 335:60–110, 2018

2018
[20]

Shmerkin

P. Shmerkin. On Furstenberg’s intersection conjecture, self -similar measures, and the Lq norms of convolutions. Ann. of Math. (2) , 189(2):319–391, 2019. 11

2019

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [5]

Dekking, K

M. Dekking, K. Simon, B. Sz´ ekely, and N. Szekeres. The interior of randomly perturbed self-similar sets on the line. Adv. Math., 448:Paper No. 109724, 43, 2024

work page 2024

[2] [1]

B´ ar´ any, K

B. B´ ar´ any, K. Simon, and B. Solomyak. Self-similar and self-aﬃne sets and measures , volume 276 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, [2023] ©2023

work page 2023

[3] [2]

V. I. Bogachev. Measure theory. Vol. I, II . Springer-Verlag, Berlin, 2007

2007

[4] [3]

Cawley and R

R. Cawley and R. D. Mauldin. Multifractal decompositions of Moran fractals. Adv. Math., 92(2):196– 236, 1992

work page 1992

[5] [4]

Cs¨ ornyei, T

M. Cs¨ ornyei, T. Jordan, M. Pollicott, D. Preiss, and B. Solomyak . Positive-measure self-similar sets without interior. Ergodic Theory Dynam. Systems , 26(3):755–758, 2006

work page 2006

[6] [6]

Erraoui and Y

M. Erraoui and Y. Hakiki. Images of fractional Brownian motion w ith deterministic drift: positive Lebesgue measure and non-empty interior. Math. Proc. Cambridge Philos. Soc. , 173(3):693–713, 2022

work page 2022

[7] [7]

J. M. Fraser and P. Shmerkin. On the dimensions of a family of over lapping self-aﬃne carpets. Ergodic Theory Dynam. Systems , 36(8):2463–2481, 2016

work page 2016

[8] [8]

Grafakos

L. Grafakos. Modern Fourier analysis , volume 250 of Graduate Texts in Mathematics . Springer, New York, second edition, 2009. 10

work page 2009