Recent progress on pointwise normality of self-similar measures

Amir Algom

REVIEW 2 major objections 6 minor 41 references

Recent progress on pointwise normality of self-similar measures

T0 review · 2 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A self-similar measure whose Fourier transform vanishes at infinity is pointwise absolutely normal, with no rate of decay needed.

desk verdict A readable, honest survey with a self-contained proof of a known theorem; the proof is sound in substance but the opening reduction to [0,1] is not justified as written. read the letter →

arxiv 2504.18192 v1 pith:MZUHNEFJ submitted 2025-04-25 math.DS math.CAmath.NT

Amir Algom This is my paper · ORCID

classification math.DSmath.CAmath.NT MSC 37A4528A8011K16

keywords self-similarmeasuresnormalnumbersRajchmanpropertyFourierdecayequidistributionmetricnumbertheoryfractalsuniformdistribution

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

This paper gives a self-contained proof of a recent theorem: if a non-atomic self-similar measure on the line has Fourier transform tending to zero at infinity — the Rajchman property — then almost every point in its support is absolutely normal, meaning normal in every integer base. This removes the quantitative decay rate required by the classical criterion from 1964, which is rarely verifiable for fractal measures even when decay is known qualitatively. The proof compares the statistics of base-$b$ orbits of typical points with statistics of conditional measures on a carefully chosen stopping-time partition, and shows that the Rajchman property alone forces the difference to vanish. The paper also explains how this theorem, together with a classification of Rajchman self-similar measures and a result on base-$b$ normality for non-commensurable contraction ratios, yields a structural description of the only obstructions to absolute normality: digit-like restrictions after an affine conjugation.

What carries the argument

The load-bearing mechanism is the orbit-comparison theorem stated as Theorem 2.1: for a compact space, a continuous map $T$, and a refining sequence of Borel partitions $A_k$ satisfying uniform diameter decay (2.2), the empirical orbit averages $\frac1N\sum_{n=1}^N \delta_{T^n x}$ and the averages of pushed conditional measures $\frac1N\sum_{n=1}^N T^n \nu_{A_n(x)}$ have the same weak-* limit for $\nu$-almost every $x$. The proof of Theorem 1.2 applies this on the space $\mathcal{A}^{\mathbb{N}}\times\mathbb{T}$, where the partitions are cut by stopping times $\tau_n(\omega)$ chosen so that the derivative of the cylinder map is roughly $b^{-n}$; condition (2.2) is verified through the uniform lower bound $\tau_n(\omega)\ge c n$ together with the uniform boundedness of the contraction ratios. This reduction turns base-$b$ normality into a Fourier statement: along the pushed measures, every nonzero integer frequency is multiplied by a factor bounded below by $c|E|$, so the Rajchman property forces the relevant Fourier coefficients to zero.

What would settle it

Construct (or find) a non-atomic self-similar measure $\nu$ with $F_\xi(\nu)\to 0$ as $|\xi|\to\infty$ but with a positive-$\nu$ set of points that fail to be $b$-normal in some integer base $b\ge2$. The theorem predicts no such measure exists, so a concrete example, or an explicit numerical computation of orbit statistics for a candidate measure showing non-equidistribution on a set of positive measure, would settle the claim.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: every non-atomic self-similar measure $\nu$ on $\mathbb{R}$ with the Rajchman property — $\widehat{\nu}(\xi)=\int e^{2\pi i \xi x}\,d\nu(x)\to 0$ as $|\xi|\to\infty$ — is pointwise absolutely normal, meaning $\nu$-almost every $x$ is normal to every integer base $b\ge 2$. The proof given here does not require any rate of decay, which is precisely the improvement over the 1964 criterion. The argument fixes a base $b$, applies the orbit-comparison theorem to the graph of the coding map over $\mathcal{A}^{\mathbb{N}}\times\mathbb{T}$, and reduces $b$-normality to vanishing of Fourier coefficients of pushed conditional measures; the Rajchman property supplies that vanishing uniformly because the pushed frequencies are bounded below by $c|E|$. From this the paper derives a structural classification: if a self-similar measure is not pointwise absolutely normal, then after an affine conjugation its IFS has contraction ratios whose logarithms are rational multiples of $\log b$ and translations of the form $k/b^q$ for some integer $b>1$, so the failure is caused by digit-like restrictions.

Load-bearing premise

The load-bearing premise is the uniform shrinking condition (2.2): the diameters of the stopping-time partition cells must tend to zero uniformly after $B$ iterates of the base-$b$ map, which the proof obtains from the uniform lower bound $\tau_n(\omega)\ge c n$ and from the contraction ratios of the IFS being bounded away from 1. If an IFS had contraction ratios accumulating at 1, this uniform estimate would break and the transfer argument would no longer go through.

Editorial extensions

If this is right

Every Rajchman self-similar measure is pointwise absolutely normal; no decay rate is needed.
If a self-similar measure is not pointwise absolutely normal, its IFS must admit an affine conjugation under which the logarithms of the contraction ratios are rational multiples of $\log b$ and the translations are of the form $k/b^q$ — digit restrictions are essentially the only obstruction.
If even one contraction ratio has logarithm not rationally commensurate with $\log b$, then almost every point of the attractor is $b$-normal.
The theorem gives a route to absolutely normal numbers inside singular fractal sets with Fourier decay, including many self-similar sets with overlaps.
Effective versions of the equidistribution remain open: the martingale argument is qualitative, so rates of convergence are not supplied.

Reading between the lines

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

The proof relies on affine structure to shift Fourier frequencies by a factor bounded below; a natural extension would test whether vanishing Fourier decay still forces normality for non-conformal IFSs, where the same frequency-shift argument needs a nonlinear analogue.
A concrete numerical test would sample points from a Rajchman self-similar measure with overlaps and estimate base-2 digit frequencies over long blocks; the theorem predicts convergence with no exceptional digits, and the observed rate would measure how far the qualitative theorem is from an effective one.
The contrapositive suggests a sharper, unproved link: inside a self-similar measure, positive measure failure of absolute normality should force the Fourier transform to have a nonzero accumulation point at infinity, and one could investigate whether the size of the exceptional set is governed by the rate of Fourier growth.

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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: this is an exposition of a recent result (Theorem 1.2) that is already published in [8], and the paper itself says so. The main value is the self-contained proof and the clean survey of the surrounding machinery. No new theorems are proved, but that's fine for an exposition if the proof is reliable. On the whole it is, with one important caveat.

What it does well: the proof of Theorem 1.2 is assembled carefully. Lemma 3.1 is a Fourier criterion for pointwise normality that is transparent. Lemma 3.2 applies Hochman's martingale theorem to the graph measure on A^N × T, using a stopping time to verify the diameter condition without any separation assumption. The final Rajchman estimate at rescaled frequencies is correct. The survey of Theorem 1.1 and Theorem 1.3, and the open problems (effective rates, non-integer bases, higher correlations), are informative and honest about what is known. The references are cited properly; the self-citation to [8] is for provenance, not to smuggle in the result.

The soft spots. The most substantial is the WLOG step at the start of Section 3: 'without loss of generality we may assume Φ preserves I=[0,1]'. This is not a harmless normalization. The theorem's conclusion is absolute normality of typical points, and absolute normality is not invariant under arbitrary real affine maps. The map that sends the invariant interval to [0,1] generally involves an irrational scaling and translation, and there is no argument that this preserves normality to all bases. The proof that follows is essentially coordinate-free and would go through verbatim for any invariant interval, so the gap is removable, but as written it is a real gap. The stress-test note about this is correct. Fixing it means either removing the WLOG and working on the original interval, or proving directly that the specific affine normalization preserves absolute normality.

Minor: Section 3.3 says 'we are now in position to prove Theorem 1.3' where it means Theorem 1.2. The inequality 'if N = N(ε)·c0' is garbled; it should be something like N(ε)/c0. These are easy fixes.

Bottom line: this is a useful survey and a readable proof of a known theorem, but the normalization gap has to be fixed before it can be taken as a rigorous self-contained proof. It deserves peer review because the expository contribution is real and the proof has enough new presentation to merit checking. I would not cite it in my own work over the original [8], but I'd send it back for revision with a request for the normalization fix, not because the theorem is in doubt.

Referee Report

2 major / 6 minor

Summary. The paper is an expository article on recent progress toward proving pointwise normality of typical points in the support of self-similar measures on the real line. Its main technical contribution is an essentially self-contained proof of Theorem 1.2: every non-atomic self-similar measure whose Fourier transform vanishes at infinity (a Rajchman measure) is pointwise absolutely normal. The proof combines Hochman's equidistribution theorem (Theorem 2.1) with a stopping-time transfer argument that reduces the problem to the Rajchman property at rescaled frequencies. The paper also explains how Theorem 1.2, together with a normality criterion from Algom--Baker--Shmerkin and Barany--Kaenmaki--Pyorala--Wu, and the recent classification of Rajchman self-similar measures by Li--Sahlsten and Bremont, yields a structural description of self-similar measures that fail to be pointwise absolutely normal (Theorem 1.1). Several open problems are discussed, concerning effective equidistribution, non-integer bases, and higher-order correlations.

Significance. If the proof is correct after the gaps noted below are fixed, Theorem 1.2 is a substantial improvement over the classical Davenport--Erdos--LeVeque criterion for self-similar measures, because it requires no quantitative decay rate on the Fourier transform. The paper gives a valuable, largely self-contained exposition of Hochman's method and makes explicit how the Rajchman property alone forces absolute normality for this class. The discussion of open problems, especially the non-integer base case and higher-order correlations, is thoughtful and well connected to recent literature. The paper also includes a complete proof of Theorem 2.1, which is a useful pedagogical contribution. These strengths make the paper potentially valuable to researchers in metric number theory and fractal geometry.

major comments (2)

[Section 3, opening paragraph] The reduction "Without the loss of generality we may assume that Φ preserves the interval I=[0,1]" is not justified by an arbitrary affine conjugacy, because pointwise absolute normality is not invariant under general real affine maps. The subsequent proof is coordinate-free and works for any invariant compact interval; please either carry a general interval I through the proof or explicitly state that the proof is written for I=[0,1] only for notational convenience and that no transfer of normality under a change of variables is being used. As written, the proof of Theorem 1.2 covers only measures with an invariant interval [0,1].
[Section 3.3, Eq. (3.7)] The sufficient condition (3.7) bounds only the single term |F_E(T_b^B ∘ φ_{ω|τ_B(ω)}_* ν)|, but Lemma 3.2 involves the Cesàro average (1/N)Σ_{n=0}^{N-1} T_b^n ∘ φ_{ω|τ_B(ω)}_* ν. To conclude the desired limsup bound, one must additionally argue that for n≥B the frequencies E b^n φ'_{ω|τ_B(ω)}(0) have magnitude at least |E|c0 so the same Rajchman estimate applies, and that the finitely many n<B terms contribute negligibly to the Cesàro average. This argument is missing. Relatedly, the statement of Lemma 3.2 should quantify B explicitly ("for every B∈N").

minor comments (6)

[Section 3.3, first sentence] The sentence "We are now in position to prove Theorem 1.3" should read "Theorem 1.2".
[Section 3.3, after Eq. (3.7)] The phrase "if R=R·c0" appears to be a typo; it should be "if R=R'/c0".
[Lemma 3.2, statement] The lemma statement uses τ_B(ω) but does not quantify B; add "for every B∈N" to the statement.
[Lemma 3.2, proof] The text says "the diameter of the projection of T^B A to the first coordinate" twice; the second occurrence should refer to the second coordinate.
[Section 1, paragraph after Theorem 1.2] The phrase "We note, however, the such bounds are usually hard to obtain" contains a grammatical error; "the such" should be "such".
[Abstract] There is a typographical spacing issue in "The orem" in the abstract; this should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.2 is derived from the exogenous Rajchman hypothesis and Hochman's equidistribution theorem; self-citations are provenance only.

full rationale

I checked the derivation chain leading to Theorem 1.2, which is the paper's main proof. Section 3.1 reduces pointwise b-normality to Fourier-smallness of empirical orbit measures (Lemma 3.1); this is a standard Weyl-criterion reformulation and does not build normality into the input. Section 3.2 applies Hochman's Theorem 2.1 to the graph measure on A^N x T; the only nontrivial verification is the diameter condition (2.2), which follows from the uniform stopping-time bound tau_B(omega) >= cB in (3.6) and the bounded contraction ratios. Section 3.3 then uses the Rajchman hypothesis exactly: after unwinding T_b^B composed with phi_{omega|tau_B} via (3.8), the Fourier coefficient of the pushed self-similar measure equals F_{xi * b^B * phi'_{omega|tau_B}(0)}(nu), and the lower bound |b^B phi'(0)| >= c0 turns the assumed decay F_xi(nu) -> 0 into the desired uniform decay (3.7). No parameter is fitted from data that includes normality, no definition of the stopping time or graph partition invokes the conclusion, and the theorem is not imported only from [8]: a full proof is supplied in the paper. The self-citations to [8] and [4] are to provenance and to the independently proved Theorem 1.3, neither of which is load-bearing for Theorem 1.2. The WLOG normalization to [0,1] would need an argument if read as an affine conjugacy, since arbitrary affine maps do not preserve absolute normality, but that is a rigor gap in coordinate choice, not a circular input: the subsequent estimates are stated for the given IFS and do not redefine normality in terms of the normalized interval. I therefore find no circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof of Theorem 1.2 uses the martingale ergodic theorem and standard self-similar measure theory; no free parameters or invented entities appear. The survey portions additionally assume two deep external theorems: the Rajchman classification and Theorem 1.3.

assumptions (5)

standard math The ergodic theorem for martingale differences (Theorem 2.2) holds for uniformly bounded sequences adapted to an increasing filtration with a fixed lag.
Invoked in the proof of Hochman's Theorem 2.1 in Section 2. The paper cites Feller [20, Chapter 7, Theorem 3] and Hochman [23, Section 2.1] instead of proving it.
standard math A finite IFS of contracting similarities has a unique nonempty compact attractor E and, for any positive probability vector p, a unique self-similar measure nu satisfying nu = sum_i p_i * phi_i#nu.
Used in Section 1 to define self-similar sets and measures. This is Hutchinson's theorem, stated as well known.
standard math For the Bernoulli measure P on the shift space, the conditional distribution on a cylinder of length m has tail independent of the prefix, so the projected conditional measure is pi composed with f_{omega|m} nu.
Used in Lemma 3.2 to identify the projection of the conditional measure of mu on a stopping-time cell. This is a property of product measures.
domain assumption The Rajchman classification of self-similar measures by Li-Sahlsten and Bremont: a self-similar measure fails to be Rajchman only if its IFS is affinely conjugate to a system with Pisot-type contraction ratios and algebraic translations.
Used in the derivation of Theorem 1.1 in Section 1.1, cited to [13,29] and not proved in this paper.
domain assumption Theorem 1.3: if some contraction ratio a_i satisfies log|a_i|/log b not in Q, then every non-atomic self-similar measure is pointwise b-normal.
Used in the proof of Theorem 1.1 to rule out the case where the Pisot base is not multiplicatively related to any integer. The theorem is cited to [4, Theorem 1.1] and [11, Theorem 1.4] and not reproved.

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Cite this review

Pith. "Pith review of Recent progress on pointwise normality of self-similar measures." pith.science (2026). https://pith.science/paper/MZUHNEFJ

@misc{pith2026250418192,
  author       = {Pith},
  title        = {Pith review of: Recent progress on pointwise normality of self-similar measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZUHNEFJ}},
  note         = {Machine review of arXiv:2504.18192}
}

read the original abstract

This article is an exposition of recent results and methods on the prevalence of normal numbers in the support of self-similar measures on the line. We also provide an essentially self-contained proof of a recent Theorem that the Rajchman property (decay of the Fourier transform) implies that typical elements in the support of the measure are normal to all bases; as no decay rate is required, this improves the classical criterion of Davenport, Erdos, and LeVeque (1964). Open problems regarding effective equidistribution, non-integer bases, and higher order correlations, are discussed.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [8]

Po intwise normality and Fourier decay for self-conformal measures

Amir Algom, Federico Rodriguez Hertz, and Zhiren Wang. Po intwise normality and Fourier decay for self-conformal measures. Adv. Math., 393:Paper No. 108096, 72, 2021

work page 2021

[2] [1]

On the pair correl ations of powers of real numbers

Christoph Aistleitner and Simon Baker. On the pair correl ations of powers of real numbers. Israel J. Math., 242(1):243–268, 2021

work page 2021

[3] [2]

Gap statistics and higher correlations for geometric progressions modulo one

Christoph Aistleitner, Simon Baker, Niclas Technau, and Nadav Y esha. Gap statistics and higher correlations for geometric progressions modulo one . Math. Ann., 385(1-2):845–861, 2023

work page 2023

[4] [3]

Pair correlations and equidis- tribution

Christoph Aistleitner, Thomas Lachmann, and Florian Pau singer. Pair correlations and equidis- tribution. J. Number Theory, 182:206–220, 2018

work page 2018

[5] [4]

On normal num bers and self-similar mea- sures

Amir Algom, Simon Baker, and Pablo Shmerkin. On normal num bers and self-similar mea- sures. Adv. Math., 399:Paper No. 108276, 17, 2022

work page 2022

[6] [5]

Van der Corput and metric theorems for geometric progressions for self-similar measures

Amir Algom, Yuanyang Chang, Meng Wu, and Yu-Liang Wu. Van d er corput and metric the- orems for geometric progressions for self-similar measures. arXiv preprint arXiv:2401.01120, 2024

work page Pith review arXiv 2024

[7] [6]

Lo garithmic Fourier decay for self conformal measures

Amir Algom, Federico Rodriguez Hertz, and Zhiren Wang. Lo garithmic Fourier decay for self conformal measures. J. Lond. Math. Soc. (2) , 106(2):1628–1661, 2022

work page 2022

[8] [7]

Po lynomial Fourier decay and a cocycle version of dolgopyat’s method for self conform al measures

Amir Algom, Federico Rodriguez Hertz, and Zhiren Wang. Po lynomial Fourier decay and a cocycle version of dolgopyat’s method for self conform al measures. arXiv preprint arXiv:2306.01275, 2023

arXiv 2023