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Metric results for dyadic approximation on the middle-third Cantor set

T0 review · 1 major / 2 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read The middle-third Cantor set has zero measure for dyadic approximations when the exponent exceeds about 1.429 and full measure below about 0.052.

desk verdict The paper improves the known thresholds on Velani's conjecture for dyadic approximation by the Cantor measure via a new uniform bound on summed squared Fourier coefficients at dyadic frequencies. read the letter →

arxiv 2606.25305 v1 pith:7LZQARL3 submitted 2026-06-24 math.NT math.DS

Xin-Rong Dai , Bing Li , Bo Wang , Yu-Feng Wu This is my paper · ORCID
classification math.NTmath.DS
keywords middle-thirdCantorsetdyadicapproximationFourierdecaymetricDiophantineself-similarmeasuresHausdorffdimensionBorel-Cantelli
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 shows that the middle-third Cantor set equipped with its natural measure has zero measure for the set of points admitting dyadic approximations of order τ when τ exceeds roughly 1.429, and full measure when τ is smaller than roughly 0.052. This partially resolves the expected transition at τ=1 by extending the known null range past 1.552 and the known full-measure range past 0.01. The proof proceeds by establishing a uniform bound on the summed squared Fourier coefficients of the measure evaluated at frequencies that are integer multiples of powers of 2, then feeding the resulting decay into Borel-Cantelli estimates that control the measure of the relevant limsup sets.

What carries the argument

The uniform bound sum from n=1 to N of |Fourier transform of the Cantor measure at h 2^n|^2 is much less than N to the power 1 minus γ, which supplies the decay needed to bound the measure of the limsup sets that define the approximation property.

What would settle it

A direct numerical check or analytic construction showing that for some τ strictly between 1 and 1.429 the measure of points with infinitely many dyadic approximations of order τ is positive, or that for some τ near 0.1 the measure is strictly less than one.

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Extended reading notes

Core claim

The authors prove that the measure of the set of points x in the Cantor set satisfying ||2^n x|| < n^{-τ} for infinitely many n is zero whenever τ exceeds 1/γ minus (1-γ)/(3-γ) and is one whenever 0<τ is less than γ/12, where γ equals log 2 over log 3. The argument rests on the new estimate that the sum from n=1 to N of the squared modulus of the Fourier transform at h times 2^n is bounded by a constant times N to the power 1 minus γ, uniformly in every nonzero integer h, together with the derived one-sided and weighted sum bounds that suffice for the measure calculations on both the null and full sides.

Load-bearing premise

The summed squared Fourier coefficients of the Cantor measure along the sequence of frequencies h times powers of 2 remain bounded by N to the power 1 minus the dimension, uniformly over all nonzero integer frequencies h.

Editorial extensions

If this is right

  • The null result now covers all τ larger than the new threshold instead of the earlier threshold near 1.552.
  • The full-measure result now reaches all τ down to the new lower threshold instead of only down to 0.01.
  • The same Fourier-sum technique yields the full-measure statement for self-similar measures supported on a broader class of missing-digit sets.
  • All three derived sum bounds hold uniformly in the auxiliary frequency parameter h.

Reading between the lines

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

  • If the Fourier-sum bound can be improved beyond the exponent 1-γ, the gap between the proven thresholds and the conjectured transition at τ=1 would shrink.
  • The same style of dyadic Fourier estimates could be tested on Diophantine approximation problems for other self-similar measures whose Fourier transforms admit comparable decay.
  • The current separation from τ=1 appears to be an artifact of the available decay rate rather than a geometric obstruction inherent to the Cantor set.
  • pith_inferences
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The paper proves partial results towards Velani's conjecture on the μ-measure of the set W_2(τ) of points in the middle-third Cantor set C that are well approximable by dyadic rationals to order τ. Specifically, it shows μ(W_2(τ))=0 for τ > 1/γ − (1−γ)/(3−γ) ≈ 1.429 and μ(W_2(τ))=1 for 0 < τ < γ/12 ≈ 0.052 (γ = log 2 / log 3), improving prior thresholds of ≈1.552 (null part) and 0.01 (full-measure part). The proofs rely on establishing the uniform Fourier estimate ∑_{n=1}^N |μ̂(h 2^n)|^2 ≪ N^{1−γ} (and its corollaries) for the Cantor-Lebesgue measure μ, which is then fed into Borel-Cantelli arguments; the method also extends to certain other self-similar measures.

Significance. If the new uniform Fourier estimates hold, the work meaningfully advances metric Diophantine approximation on self-similar sets by narrowing the gap to the full conjecture and supplying a technical tool (uniform decay of Fourier sums along dyadic orbits) that may apply more broadly. The explicit improvement over Allen-Baker-Chow-Yu (2023) and Baker (2025) is concrete, and the generalization statement is a positive feature.

major comments (1)
  1. [Abstract (key estimate) and the section deriving the Fourier bounds] The central claims rest on the derivation of the estimate ∑_{n=1}^N |μ̂(h 2^n)|^2 ≪ N^{1−γ} (uniform in h ≠ 0) and its two corollaries; the abstract states these are proved in the paper and then applied via standard Borel-Cantelli arguments, but the uniformity in h and the precise range of σ in the weighted sum must be verified in the body to confirm they are load-bearing for both the null and full-measure statements.
minor comments (2)
  1. [Abstract and introduction] The numerical approximations 1.429 and 0.052 should be accompanied by the exact algebraic expressions throughout for precision.
  2. [Preliminaries] Notation for the Fourier transform μ̂ should be defined explicitly on first use, including the normalization convention.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract (key estimate) and the section deriving the Fourier bounds] The central claims rest on the derivation of the estimate ∑_{n=1}^N |μ̂(h 2^n)|^2 ≪ N^{1−γ} (uniform in h ≠ 0) and its two corollaries; the abstract states these are proved in the paper and then applied via standard Borel-Cantelli arguments, but the uniformity in h and the precise range of σ in the weighted sum must be verified in the body to confirm they are load-bearing for both the null and full-measure statements.

    Authors: The uniformity in h ∈ ℤ mid {0} and the range 0 < σ < 1 − γ/2 are established explicitly in the proofs of the key estimates (Theorem 1.2 and its corollaries) in Section 3. The derivation proceeds via a uniform bound on the Fourier coefficients along dyadic orbits that holds independently of h, with the weighted sum following by a standard summation-by-parts argument that preserves the uniformity. These estimates are then invoked directly in the Borel–Cantelli arguments of Sections 5 (null part) and 6 (full-measure part), so they are load-bearing for both statements as claimed. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation self-contained; no circular reductions

full rationale

The paper derives the key uniform Fourier estimates ∑_{n=1}^N |μ̂(h 2^n)|^2 ≪ N^{1-γ} (and corollaries) as an independent technical contribution, then applies them directly in Borel-Cantelli arguments to obtain the metric statements for τ in the stated ranges. These estimates are not obtained by fitting to the target limsup sets, by self-citation chains, or by renaming prior results; they are presented as newly proved and uniform in h. Prior improvements cited are from distinct authors (Allen-Baker-Chow-Yu 2023; Baker 2025), and the argument contains no self-definitional steps or ansatz smuggling. The derivation is therefore self-contained against external benchmarks.

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

The proof rests on standard Fourier analysis and measure-theoretic Borel-Cantelli lemmas together with the new sum estimate whose derivation is internal to the paper; no free parameters or invented entities are introduced.

assumptions (2)
  • standard math Standard properties of the Fourier transform of a probability measure and the definition of the middle-third Cantor set and its measure
    Invoked throughout the derivation of the summed Fourier coefficient bounds
  • standard math Borel-Cantelli lemmas for the measure of limsup sets
    Used to translate the Fourier bounds into statements about μ(W_2(τ))

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Cite this review

Pith. "Pith review of Metric results for dyadic approximation on the middle-third Cantor set." pith.science (2026). https://pith.science/paper/7LZQARL3

@misc{pith2026260625305,
  author       = {Pith},
  title        = {Pith review of: Metric results for dyadic approximation on the middle-third Cantor set},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7LZQARL3}},
  note         = {Machine review of arXiv:2606.25305}
}
abstract

Let $C$ be the middle-third Cantor set and $\mu$ be the Cantor-Lebesgue measure on $C$. A conjecture of Velani states that $\mu(W_2(\tau))=0$ if $\tau>1$ and $\mu(W_2(\tau))=1$ if $0<\tau\leq 1$, where $W_2(\tau)=\left\{x\in[0,1]: \|2^nx\|<n^{-\tau}\ {\rm for\ infinitely\ many }\ n\in\mathbb{N} \right\}$. We prove that the conjecture holds for $\tau>\frac{1}{\gamma}-\frac{1-\gamma}{3-\gamma}\,(\approx 1.429)$ and $0<\tau<\frac{\gamma}{12}\,(\approx 0.052)$, where $\gamma=\frac{\log2}{\log3}$ is the Hausdorff dimension of $C$. This improves the known results on both the null part ($\tau>\frac{1}{\gamma}-\frac{0.078(1-\gamma)}{\gamma(2-\gamma)}\approx 1.552$, due to Allen, Baker, Chow, and Yu (2023)) and the full measure part ($0<\tau\leq 0.01$, due to Baker (2025)). Our key innovation is to establish the estimate \[\sum_{n=1}^{N}|\widehat{\mu}(h2^n)|^2\ll N^{1-\gamma}\] and its consequences: \[ \sum_{n=1}^{N}|\widehat{\mu}(h2^n)|\ll N^{1-\frac{\gamma}{2}},\quad \sum_{n=1}^{N}n^{- \sigma}|\widehat{\mu}(h2^n)|\ll_{\sigma} N^{1-\frac{\gamma}{2}-\sigma},\] where $0<\sigma<1-\frac{\gamma}{2}$, and all estimates are uniform in $h\in\mathbb{Z}\setminus\{0\}$. For the full measure part, our approach also generalizes to self-similar measures on a class of missing-digit sets.

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Works this paper leans on

24 extracted references · 2 canonical work pages

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