Explicit Upper Bounds on Decay Rates of Fourier Transforms of Self-similar Measures on Self-similar Sets

Ying Wai Lee

REVIEW 1 major objections 4 minor 1 cited by

Explicit Upper Bounds on Decay Rates of Fourier Transforms of Self-similar Measures on Self-similar Sets

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Pith's one-line read Every self-similar measure on a suitably arithmetic self-similar set gets an explicit logarithmic Fourier decay rate.

desk verdict A careful, honest paper that makes Li-Sahlsten's logarithmic decay exponent explicit; the main proof structure is sound, but Proposition 3 and Proposition 9 need fixing before publication. read the letter →

arxiv 2412.16621 v1 pith:VLHNDHWY submitted 2024-12-21 math.CA math.NT

Ying Wai Lee This is my paper · ORCID

classification math.CAmath.NT MSC 28A8042A3860K0511K55

keywords Fouriertransformsofmeasuresself-similarsetsRajchmanpropertyquantitativerenewaltheorynon-LiouvillenumbersL\"urothexpansionupperregularityexponent

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

For any self-similar measure on a self-similar set whose contraction ratios satisfy a non-Liouville condition, the Fourier transform decays like a power of $\log|\xi|$, with the exponent explicitly given in terms of the measure's upper regularity exponent and the degree of the non-Liouville ratio. The paper thereby upgrades an earlier result whose decay exponent was implicit and unusable for quantitative applications. This matters because metric number theory sometimes requires checking that a decay exponent exceeds a concrete threshold, which was impossible before. The paper also constructs a self-similar measure attaining the fastest exponent available under the open set condition, and verifies the arithmetic hypothesis for sets of numbers with digits restricted to a finite set in their L\"uroth representation.

What carries the argument

The load-bearing mechanism is a quantitative renewal theorem for the stopping time of a random walk whose increments are $-\log r_i$ with probabilities $p_i$. The auxiliary step distribution must be weakly diophantine, meaning $|t|^{2s-2}|1-\mathcal{L}\lambda(it)|$ stays bounded away from zero; Proposition 3 derives this from the non-Liouville condition on $\log r_i/\log r_j$. Proposition 4, a renewal estimate with explicit error $O(t^{-1/(8s-7)})$, then controls the off-diagonal part of the two-dimensional integral in Proposition 1, while the diagonal part is bounded using the upper regularity exponent $\alpha$. Balancing the two terms at a scale $t$ coupled to $|\xi|$ by $t^{(1+\alpha)/((1+2\alpha)(8s-7))} e^t = |\xi|$ produces the final logarithmic exponent.

What would settle it

Take the L\"uroth set with digits $\{2,3\}$ and its optimal self-similar measure $\mu_3$; compute $|\hat{\mu}_3(\xi)|$ numerically at frequencies $\xi=e^n$ for large $n$ and compare the envelope with $C\log^{-\beta_3}\xi$ for the paper's $\beta_3>10^{-11}$. If the measured decay along this sequence is slower than every positive power of $\log|\xi|$, or if the envelope exceeds the predicted bound by an unbounded factor, the claimed universal rate would be refuted.

Watch

Extended reading notes

Core claim

Theorem 1 is the paper's central assertion: if $\log r_i/\log r_j$ is non-Liouville of degree $s\ge 2$ for some two similitudes of a self-similar set $E$, then every self-similar probability measure $\mu$ on $E$ satisfies $|\hat{\mu}(\xi)|=O(\log^{-\alpha/(2(1+2\alpha)(8s-7))}|\xi|)$ as $\xi\to\pm\infty$, for every upper regularity exponent $\alpha$ of $\mu$. The proof splits the Fourier integral into a diagonal region controlled by $\alpha$ and an off-diagonal region controlled by a quantitative renewal estimate for the random walk with step distribution given by the logarithmic contraction ratios. Under the open set condition, the paper's Theorem 3 shows that the measure with weights $r_i^{\dim E}$ has $\alpha=\dim E$, so the fastest rate available from this method is $O(\log^{-\dim E/(2(1+2\dim E)(8s-7))}|\xi|)$. For restricted L\"uroth digit sets the needed non-Liouville hypothesis is checked by an explicit lower bound on linear forms in logarithms, and the resulting positive decay exponent is spelled out.

Load-bearing premise

The load-bearing premise is that the logarithm of one contraction ratio divided by the logarithm of another is a non-Liouville number of some finite degree; if that fails, the auxiliary measure is not provably weakly diophantine and the quantitative renewal estimate collapses.

Editorial extensions

If this is right

Every self-similar measure on such a set has a computable, positive logarithmic decay rate, so the earlier implicit parameter becomes an explicit function of the measure and the set.
Under the open set condition, the optimal measure is the one with weights $p_i = r_i^{\dim E}$, and its decay exponent is $\dim E/(2(1+2\dim E)(8s-7))$, with $\dim E$ the unique solution of $\sum_i r_i^q=1$.
For finite restricted L\"uroth digit sets, there is an explicit self-similar measure with decay like $\log^{-\beta}|\xi|$, where $\beta$ is written in terms of the two smallest digits; for digits $\{2,3\}$, $\beta>10^{-11}$.
Because the decay is logarithmic with positive exponent, the measure satisfies the summability condition in the classical uniform-distribution criterion, so for $\mu$-almost every $x$ any lacunary sequence $(n_k x \bmod 1)$ is uniformly distributed.
The explicit exponent still falls short of the threshold $\beta>2$ needed in the stated inhomogeneous Diophantine approximation application; the paper's own best case gives $\beta=1/54$.

Reading between the lines

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

If the non-Liouville hypothesis is dropped, the renewal estimate loses its handle; it is an editorial guess that some self-similar measures with a Liouville log-ratio will exhibit no positive logarithmic decay, and exhibiting such an example would mark the true boundary of the theorem.
The explicit constants are so small in the L\"uroth instance that the practical route to thresholds like $\beta>2$ is likely to run through sharper Baker-type bounds and sharper renewal error terms, rather than through the present architecture.
The same template should transfer to other self-similar digit systems---continued-fraction Cantor sets, $\beta$-expansions, and other L\"uroth-type bases---wherever the contraction ratios are algebraic and explicit lower bounds for linear forms in logarithms are available.
The role of the open set condition seems to enter only through the regularity exponent $\alpha$; the Fourier estimate itself is independent of separation, so a different way of establishing a large $\alpha$ could produce fast decay without the separation condition.

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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 a solid, honest paper that does exactly what the title says--makes the decay exponent in Li-Sahlsten's theorem explicit. The main result, Theorem 1, gives a log^{-alpha/(2(1+2alpha)(8s-7))} bound, and the proof is mostly careful parameter-tuning of the Li-Sahlsten renewal machinery rather than a new mechanism. That is fine; the paper says so.

The genuinely new pieces: the explicit formula, the construction in Theorem 3 of a measure achieving the exponent with alpha = dim E, and the Luroth application in Theorem 4. I checked the algebra in Propositions 2 and 5 and it is consistent; the choice of t in (21) gives the claimed exponent. The paper is transparent about Moran's theorem, and the new proof of Moran's formula using the measure nu_alpha is clean. No fitted parameters, no circularity. Credit where it is due: this is an honest incremental advance.

Soft spots, in decreasing order of seriousness. First, Proposition 3 only verifies weak diophantineness on the arithmetic progression (2pi/log r_b)Z. The definition needs the liminf over all real tau. The missing perturbation argument--bounding the off-lattice frequencies by combining the non-Liouville distance with the epsilon^2 contribution from the b-term--should work, but it is not in the text. As written the proof has a genuine gap, so the theorem is conditional on that being filled. Second, Proposition 9's use of Matveev's theorem does not read cleanly: the displayed computation appears to gain an extra factor in the exponent, and the numerical constants are not re-derivable from the cited theorem as written. This deserves a careful check. Third, and minor: the exponents are extremely small (best case 1/54), and the paper honestly notes it does not reach the PVZZ threshold. That limits the payoff but is not a flaw.

The paper is for people working on Fourier decay of self-similar measures and quantitative metric number theory. It deserves a serious referee. I would send it out, with requests to fix Proposition 3 and double-check Proposition 9 against Matveev; the main argument is likely sound.

Referee Report

1 major / 4 minor

Summary. The paper proves an explicit logarithmic decay rate for Fourier transforms of self-similar probability measures on self-similar sets in [0,1], assuming that the ratio of two contraction logarithms is non-Liouville of degree s. Theorem 1 states that for any upper regularity exponent alpha of the measure, |mu-hat(xi)| = O(log^{-alpha/(2(1+2alpha)(8s-7))}|xi|) as |xi| tends to infinity. The proof combines a diagonal estimate using upper regularity with an off-diagonal estimate obtained from a quantitative renewal theorem, after verifying that an auxiliary measure is weakly diophantine. Theorem 2 gives a new proof of Moran's dimension formula, Theorem 3 constructs a self-similar measure that attains the fastest decay rate allowed by Theorem 1 under an open-set-type condition, and Theorem 4 applies the machinery to sets of numbers with digits restricted in their Luroth expansions, using Matveev's explicit Baker theorem to verify the non-Liouville condition.

Significance. If the proof is completed, the paper makes a genuine contribution: it converts the implicit logarithmic decay rate of Li-Sahlsten into an explicit, parameter-free exponent depending only on the upper regularity exponent and the non-Liouville degree. The construction in Theorem 3 of a measure achieving the maximal rate is a useful and nontrivial addition, and the application to restricted Luroth digits gives a fully explicit, though numerically small, decay exponent. The new proof of Moran's formula via upper regularity exponents is a legitimate alternative route. The main claims are quantitative and falsifiable, and the paper does not use fitted parameters or circular reasoning; its dependence on prior results by Li-Sahlsten, Li, and Matveev is clearly stated.

major comments (1)

[Section 2, Proposition 4 and its use in Proposition 5] Proposition 4 is stated for C^1 functions f : R -> [0,+infty), but in Proposition 5 it is applied to the complex-valued function f_theta(s) = exp(-2 pi i theta e^{-s}) (equation (23)). The renewal estimate is linear, so the application can be repaired by splitting into real and imaginary parts or by stating a complex-valued version, but as written the theorem statement does not cover the case actually used.

minor comments (4)

[Section 2, proof of Proposition 3] In the last paragraph of the proof, the formula for tau changes from 2 pi tau_1 / log r_b to 2 pi tau_1 / log r_a; the two denominators should be reconciled.
[Section 2, Proposition 5] The sentence defining t as the unique solution to equation (21) would benefit from an explicit remark that existence and uniqueness hold for all sufficiently large |xi|; this is implicit in the proof but not stated.
[Section 5, Theorem 4 and Proposition 10] The passage from beta_I,0 to the simpler expression beta_I uses the assertion 'beta_I,0 >= beta_I'; a one-line verification of this inequality would help the reader confirm the numerical claim.
[Section 1, Theorem 1] The statement allows the degenerate case alpha = 0, which gives only the trivial O(1) bound; a brief remark that this is consistent with the Frostman lemma and the singleton case would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the explicit decay rate is derived from stated hypotheses via external renewal-theoretic results, with no fitted parameters renamed as predictions.

full rationale

The paper's central derivation is self-contained against its stated assumptions. Theorem 1 is proved by combining a stopping-time decomposition (Proposition 1), an upper-regularity bound for the near-diagonal contribution (Proposition 2), a weak-diophantine property of the auxiliary measure (Proposition 3), and a quantitative renewal estimate (Proposition 4). None of these inputs assumes the conclusion: the target logarithmic decay rate appears only after combining the estimates, and the exponent depends explicitly on the upper regularity exponent and the non-Liouville degree. There are no fitted parameters and no use of the desired decay as an input. The cited external results [7, 8, 11] are prior theorems imported as tools, not restatements of the goal, and none is by the present author, so there is no load-bearing self-citation. Theorem 3 follows from Theorem 1 by constructing a measure whose upper regularity exponent is the Hausdorff dimension via Proposition 6 and Theorem 2; this is a genuine deduction, not a renaming. Theorem 4 is an application of Theorem 3 and Matveev's explicit lower bound, again without assuming the conclusion. The only flagged issue is a possible proof gap in Proposition 3: the weak-diophantine estimate is verified only on the lattice tau = 2 pi Z / log r_b rather than for all real tau, which may be a correctness concern for the renewal estimate. That is not circularity, because it does not make the conclusion an input; it would be a missing justification in the proof chain. Overall, no circular step is present, and the score reflects the absence of circular structure rather than an assessment of the proof gap.

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

The central claims rest on standard IFS and Hutchinson machinery, Frostman's lemma, two external analytic lemmas from the Fourier-decay literature, and one explicit Diophantine lower bound by Matveev. There are no fitted parameters or newly invented objects; the measure μ_I in Theorems 3 and 4 is a standard self-similar measure with weights adapted to the similarity dimension, and the constants in Theorem 4 are deliberately simplified.

assumptions (5)

standard math Hutchinson's theorem: for a finite IFS of similitudes there is a unique non-empty compact self-similar set and, for any positive weights summing to one, a unique self-similar probability measure.
Used in the definitions and existence statements in Section 1; cited as [5, Theorem 3.1.(3)(i)] and [5, Theorem 4.4.(1)(i)].
standard math Frostman's lemma: any upper regularity exponent α of a probability measure on E satisfies α ≤ dim E.
Used after Theorem 1 to justify that α ≤ dim E and that the constructed measure in Theorem 3 has maximal exponent; cited as [10, Theorem 8.8].
domain assumption Feng-Lau result: non-singleton self-similar measures admit a positive upper regularity exponent.
Used in Section 1 to guarantee that Theorem 1 gives nontrivial decay outside singleton cases; cited as [2, Proposition 2.2].
standard math The quantitative renewal theorem of Li-Sahlsten (Proposition 4 here) and the oscillatory integral estimate of Li (Lemma 3.8) both hold as stated for finite-support, non-lattice, weakly diophantine measures.
These are imported black boxes in Section 2; Proposition 4 is explicitly [8, Proposition 2.2] and the oscillatory estimate is used in Proposition 5 via [7, Lemma 3.8].
standard math Matveev's explicit lower bound for linear forms in two logarithms has the constants claimed in Proposition 9.
Proposition 9's non-Liouville degree bound is derived by substituting e1(e1-1) and e2(e2-1) into [11, Theorem 2.1]; the numerical values 387072, 15.8+5.5 log 2 and the factor e^3 are imported from that theorem.

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Cite this review

Pith. "Pith review of Explicit Upper Bounds on Decay Rates of Fourier Transforms of Self-similar Measures on Self-similar Sets." pith.science (2026). https://pith.science/paper/VLHNDHWY

@misc{pith2026241216621,
  author       = {Pith},
  title        = {Pith review of: Explicit Upper Bounds on Decay Rates of Fourier Transforms of Self-similar Measures on Self-similar Sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLHNDHWY}},
  note         = {Machine review of arXiv:2412.16621}
}

read the original abstract

The study of Fourier transforms of probability measures on fractal sets plays an important role in recent research. Faster decay rates are known to yield enhanced results in areas such as metric number theory. This paper focuses on self-similar probability measures defined on self-similar sets. Explicit upper bounds are derived for their decay rates, improving upon prior research. These findings are illustrated with an application to sets of numbers whose digits in their L\"uroth representations are restricted to a finite set.

Figures

Figures reproduced from arXiv: 2412.16621 by the authors.

**Figure 1.** illustrates the fractal construction process of 3 ∶= {2,3}. Starting from the interval (0, 1], sub-intervals are partitioned and removed iteratively at each level. At the first level, (0, 1] is partitioned to three sub-intervals: (0, 1/3], (1/3, 1/2], and (1/2, 1], which correspond to numbers in (0, 1] with 1 ≥ 4, 1 = 3, and 1 = 2 respectively. The first sub-interval is removed, while the second and third are retain… view at source ↗

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Pith tools

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

[1] [1]

Introduction & Results 1

work page

[2] [2]

Proof of Theorem 1 7

work page

[3] [3]

Proof of Theorem 2 12

work page

[4] [4]

Proof of Theorem 3 14

work page

[5] [5]

Proof of Theorem 4 14 References 16

work page

[6] [6]

It is later generalise d to measures, enabling the frequency analysis of distributions

I/n.sc/t.sc/r.sc/o.sc/d.sc/u.sc/c.sc/t.sc/i.sc/o.sc/n.sc & R/e.sc/s.sc/u.sc/l.sc/t.sc/s.sc The Fourier transform, a fundamental tool in harmonic analysis, is initiall y deﬁned for functions to analyse their frequency components. It is later generalise d to measures, enabling the frequency analysis of distributions. Let )u1D439 ⊂ [0, 1]and )u1D707be a prob...

work page 2024

[7] [7]

A few deﬁnitions and propositions are presented before starting the main proof of Theorem 1

P/r.sc/o.sc/o.sc/f.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc 1 The proof of Theorem 1 employs quantitative renewal theory, spec iﬁcally for the stopping time of random walks, as introduced in prior work [8]. A few deﬁnitions and propositions are presented before starting the main proof of Theorem 1. The self-similar measure /u1D707, deﬁned as in equat...

work page

[8] [8]

Let  be a non-empty ﬁnite set

P/r.sc/o.sc/o.sc/f.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc 2 A few deﬁnitions and propositions are presented before starting the main p roof of Theo- rem 2. Let  be a non-empty ﬁnite set. Deﬁne /u1D453 ∶ [0, 1] → ℝ+ by, for any /u1D460∈ [0, 1], /u1D453(/u1D460) ∶= )summation(size1 /u1D464∈ /u1D45F/u1D464 /u1D460. (26) Let /u1D460∈ [0, 1]. Deﬁne,...

work page