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REVIEW 3 major objections 5 minor 58 references

Simultaneous and multiplicative Diophantine approximation on missing-digit fractals

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Pith's one-line read A Fourier dimension threshold makes measures Khinchin and Gallagher.

desk verdict A genuinely new Fourier-dimension criterion for Khinchin and Gallagher, with a solid self-contained convergence theory, but the divergent half is conditional on an unproved second-moment estimate imported from the authors' unpublished preprint. read the letter →

arxiv 2412.12070 v2 pith:67SZEHHY submitted 2024-12-16 math.NT math.DS

Sam Chow , Han Yu This is my paper · ORCID
classification math.NTmath.DS MSC 11J8328A8042B05
keywords Diophantineapproximationmissing-digitfractalsKhinchin'stheoremGallagher'sFourierℓ1dimensionmetricnumbertheoryLittlewood'sconjecturerationalpointson
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 proves that a single number attached to a probability measure — its Fourier $\ell_1$ dimension, measuring how fast Fourier coefficients decay on average — controls whether the measure inherits the two central theorems of metric Diophantine approximation. If the dimension exceeds $k - (k-1)/(k+1)$, the measure is Khinchin: typical points are approximated by rationals at the same rate as Lebesgue-typical points, for every non-increasing approximation function. If the measure is a product of one-dimensional measures and each factor has dimension above $1 - 1/(k+1)$, it is also Gallagher, so the multiplicative version of the dichotomy holds. These give Fourier-dimension criteria that yield both properties for arbitrary Borel measures, and the appendix shows that natural missing-digit fractals — Cantor-type sets obtained by deleting digits in a large base — meet the thresholds. The reader should care because this settles, for a concrete family of fractal measures, a piece of the conjecture that natural measures behave like Lebesgue measure unless there is an obvious affine obstruction.

What carries the argument

The load-bearing objects are the Fourier $\ell_1$ dimension $\dim_{\ell_1}(\mu)$, defined as the supremum of $s$ such that the $\ell^1$ sums of Fourier coefficients over boxes of radius $Q$ grow at most like $Q^{k-s}$, and the moment transference principles. The expectation transference principle (ETP) compares $\mu$-averages of smoothed indicators of the approximation sets with their Lebesgue averages; the variance transference principle (VTP) shows the $\mu$-variance matches the Lebesgue variance up to a negligible error. The VTP is the heart of the divergence theory: after expanding in Fourier series, the non-diagonal contribution becomes a sum over integer solutions to the linear system $t n + t' n' = x$ with divisibility constraints inherited from the $n^{-1}$-periodicity of the level sets, and counting these solutions with a divisor bound produces an error controlled by $\dim_{\ell_1}(\mu)$. The threshold (1.1) is exactly the condition that this error is smaller than the square of the expectation. The admissible set and function systems decompose the simultaneous and multiplicative approximation sets into unions of rectangles $A_n(d_{i,1},\ldots,d_{i,k})$ of controlled sizes and volumes, which is what makes the Fourier analysis tractable.

What would settle it

Compute the optimal constant $C$ in (2.5) for the admissible function systems of Examples 2.5 and 2.6; if the infimum of $C$ over allowed bump functions is strictly greater than 1, then Lemma 2.4 cannot deliver $\mu(W)=1$ and the divergence conclusions of Theorem 1.13 collapse to a lower bound. Alternatively, search for a Borel probability measure with $\dim_{\ell_1}(\mu) > k - (k-1)/(k+1)$ whose Khinchin divergence set has measure strictly less than 1, which would directly refute Theorem 1.8.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1.8: any Borel probability measure $\mu$ on $\mathbb{R}^k$ with Fourier $\ell_1$ dimension $\dim_{\ell_1}(\mu) > k - (k-1)/(k+1)$ is Khinchin — the set of vectors $x$ satisfying $\max_j \|n x_j\| < \psi(n)$ for infinitely many $n$ has $\mu$-measure $1$ whenever $\sum_n \psi(n)^k = \infty$ and measure $0$ when this series converges. If in addition $\mu = \mu_1 \times \cdots \times \mu_k$ is split and each factor satisfies $\dim_{\ell_1}(\mu_j) > 1 - 1/(k+1)$, then the same dichotomy holds for the multiplicative condition $\|n x_1\|\cdots\|n x_k\| < \psi(n)$, so $\mu$ is Gallagher. The proofs establish both properties in the stronger inhomogeneous form with arbitrary shifts $y \in \mathbb{R}^k$: a convergence theory requiring only the weaker Fourier condition (1.3), and a divergence theory using (1.1). For missing-digit fractals whose digit set is a Cartesian product or nearly the full digit set, Theorem A.1 shows that $\dim_{\ell_1}$ is arbitrarily close to the Hausdorff dimension for large base $b$, so the thresholds are met; the same Fourier machinery yields counting estimates for rational points near such fractals (Theorems 1.17 and 1.20) and a refinement of intrinsic Diophantine approximation (Theorem 1.22).

Load-bearing premise

The divergence half of the main theorem rests on a Lebesgue second-moment estimate (Theorem 2.8) quoted without proof from an earlier preprint by the authors, and the proof needs the constant in that estimate to be arbitrarily close to 1 for the specific approximating functions used; if that fails, the full-measure conclusion degrades to $\mu(W) \ge 1/C$.

Editorial extensions

If this is right

  • Missing-digit measures with a product-like digit set and sufficiently large base $b$ satisfy both the Khinchin and Gallagher laws; for digit sets of size $b^k - 1$ the Fourier $\ell_1$ dimension exceeds $k - \varepsilon$ for large $b$, so the main threshold is met.
  • The inhomogeneous forms hold for every shift $y$: the sets $W_k(\psi,y)$ and $W^\times_k(\psi,y)$ have $\mu$-measure $0$ or $1$ exactly according to the convergence or divergence of their defining series, so the approximation behaviour is stable under translation of the targets.
  • For missing-digit fractals satisfying (1.3), the estimate $\#N_K(Q,\delta) \asymp \delta^{k-\dim_H(K)} Q^{\dim_H(K)+1}$ holds for $\delta \gg Q^{-\eta-1/k}$, giving power-saving bounds on rational points on the fractal and sharp bounds in extremal examples.
  • Almost every point of such a fractal is not intrinsically very well approximable for exponents $\tau > 1/k - \varepsilon$ (Theorem 1.22), refining the previous threshold.

Reading between the lines

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

  • If the thresholds (1.1) and (1.2) are sharp, then the Fourier $\ell_1$ dimension is the correct single-number statistic for metric Diophantine approximation on self-similar fractals; constructing measures just below the threshold that fail to be Khinchin or Gallagher would be a direct test of that sharpness.
  • The moment transference machinery should extend to other self-similar measures with rational contraction ratios and the open set condition, where $\dim_{\ell_1}$ can be computed by the algorithm in Theorem A.2; one could then try to push the thresholds toward the Hausdorff dimension, which is what the full Dream Theorem would require without the large-base restriction.
  • The counting results suggest a fractal analogue of dimension growth: for missing-digit fractals that do not contain affine hyperplanes, $\#N_K(Q,0)$ should be substantially smaller than the trivial $Q^k$ bound; Theorem 1.20 shows the obstruction is real and concentrated on affine-linear structure.
  • A numerical experiment for $k=2$ could test the Gallagher criterion near the boundary: computing the second-moment constant in (2.5) for truncated digit sets would indicate whether the condition $\dim_{\ell_1}(\mu_j) > 2/3$ is genuinely the right factor-level threshold.
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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

3 major / 5 minor

Summary. The paper proves metric Diophantine approximation results for Borel probability measures in terms of the Fourier ℓ1 dimension. The main theorem (Theorem 1.8) states that if dim_{ℓ1}(μ) > k - (k-1)/(k+1), then μ is Khinchin, and if, in addition, μ is a product of measures each with dim_{ℓ1} > 1 - 1/(k+1), then μ is Gallagher. The proof develops expectation and variance transference principles: the ETP uses Fourier decay of the measure in dyadic annuli, and the VTP reduces the variance to a counting estimate for solutions of a system of Diophantine equations. The authors then apply these principles to missing-digit fractals, obtaining also results on rational point counting (Theorem 1.17), Hausdorff dimension bounds (Theorem 1.15), and intrinsic approximation (Theorem 1.22).

Significance. If correct, this is the first Fourier-dimension threshold that guarantees both the Khinchin and Gallagher properties for arbitrary measures. For missing-digit fractals, Appendix A shows the thresholds are met for suitable digit sets, e.g., when the base b is large and #D = b^k - 1, so the theorem goes beyond the recent Hausdorff-dimension threshold of Khalil–Luethi. The proofs in Sections 3–5 are detailed, and the final inequality in the VTP reduces exactly to the assumed condition (1.1). The paper also provides a general algorithm to estimate dim_{ℓ1} for missing-digit measures.

major comments (3)
  1. [Section 2.6, Theorem 2.8] The divergent half of the main theorem is not self-contained. Theorem 2.8 quotes the Lebesgue second-moment estimate (2.5) with C = C(k,w) arbitrarily close to 1 from the authors' preprint [25], without proof. This estimate is load-bearing: Lemma 2.4 concludes μ(E∞) ≥ 1/C, and the full-measure statement μ(E∞) = 1 in Theorem 1.13 requires C to be arbitrarily close to 1 for the exact admissible function systems of Examples 2.5 and 2.6. If (2.5) is only available with a fixed C > 1, the divergence conclusions of Theorem 1.8 would reduce to positive-measure results. The authors should either include a proof of (2.5), or state Theorem 1.13 as conditional on the resolution of [25].
  2. [Appendix A, Theorem A.1] The paper's advertised applicability to missing-digit fractals rests on the Fourier ℓ1 dimension estimates quoted from the preprints [58, Theorem 2.15] and [23, Theorem 2.6]. These are not reproved in the present paper. Since the examples are used to show that the condition (1.1) is non-vacuous (e.g., Theorem 1.20 and the sentence following Corollary 1.18), the authors should provide the necessary statements in full or prove the needed cases.
  3. [Section 4.3] In the concluding step of the VTP proof, the notation dim_{ℓ1}(μ) is used as if the defining Fourier estimate holds at the supremum. Since dim_{ℓ1} is a supremum, the argument should fix an exponent κ slightly below dim_{ℓ1}(μ), use the corresponding estimate, and pass to the limit; the strict inequality in (1.1) leaves the required room. This is a presentation gap, but it should be corrected to make the proof fully rigorous.
minor comments (5)
  1. [Section 5.1] For the convergent Khinchin case, the replacement of ψ by max{ψ, ψ_L} is justified, but the monotonicity of the resulting function should be stated explicitly, since the classical theorems require ψ non-increasing.
  2. [Section 3.2] The statement of Theorem 3.2 uses a fixed d but allows the individual dj to vary within N^{-τ}d and N^{τ}d; the final bound is stated in terms of d. A short note on the uniformity of the implied constants in τ would be helpful.
  3. [Section 5.4] In the lower bound of Theorem 1.17, the single dyadic block [N,2N) with N = floor((Q-1)/2) is used; the wording 'whenever Q is sufficiently large' should clarify that the implied constants may depend on the measure μ through κ and κ2.
  4. [Page 5, Definition 1.11] The inhomogeneous definitions use the same symbol W_k(ψ,y) as the homogeneous case; consider adding a remark that the homogeneous results correspond to y=0.
  5. [Example 1.19] The existence of a missing-digit fractal with dim_H(K) < k - k/(k+1) and D ⊇ {0,...,b-1}^{k-1} × {0} is asserted but not demonstrated; the reader can fill this in by taking #D = b^k - 1 with large b, but a one-line explanation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ℓ1-dimension condition is a hypothesis, and the imported Lebesgue second-moment estimate from [25] is independent of the target result.

full rationale

The derivation chain is not circular. Theorem 1.8 is conditional on the Fourier ℓ1-dimension thresholds (1.1) and (1.2), which are independent hypotheses rather than conclusions; the expectation and variance transference principles (Theorems 3.1, 3.2, 4.1) are proved in the present paper from those hypotheses, not imported as the target. The convergence half (Theorem 1.12) is self-contained modulo Lemma 2.3, a standard Borel–Cantelli packaging lemma adapted from [25]. The divergence half (Theorem 1.13) invokes Theorem 2.8 from [25] for the Lebesgue second-moment bound (2.5), but that estimate concerns Lebesgue measure and the chosen bump-function system, with no ℓ1-dimension or Khinchin/Gallagher assumption, so it is an independent supporting input rather than a restatement of the main theorem. The Appendix A examples use Theorem A.1 from [58], which is again a parameter-free computation of dim_ℓ1 for missing-digit measures, not an assumption of the conclusion. The self-citations are procedural; none defines the main theorem in terms of itself, and the only substantial concern is that the quoted estimate from the unpublished preprint [25] is not reproved here, which is a completeness or correctness risk outside the circularity framework.

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

The central claim depends only on standard Fourier analysis, the Borel-Cantelli lemmas, and two imported technical estimates: the Lebesgue second-moment bound from [25] (Theorem 2.8) and the l1-dimension computations from [58] (Appendix A.1) used to build examples. No free parameters are fitted to data; the proof constants (epsilon, tau, kappa, eta) are chosen universally. No new entities are postulated.

assumptions (5)
  • standard math Divergence Borel-Cantelli lemma (Beresnevich-Velani [13])
    Used in Lemma 2.4 to convert ETP/VTP plus divergence of lambda(f_n) into mu(W) >= 1/C, and then to full measure via an exhaustion argument; cited in section 2.3.
  • standard math Fourier inversion and Parseval on the torus for Schwartz functions
    Used throughout sections 3 and 4 to express ETP and VTP integrals in terms of Fourier coefficients; functions are Schwartz so the series converge absolutely, as stated in section 4.
  • domain assumption Lebesgue second-moment estimate (2.5) from [25, Theorem 2.8]
    Imported from the authors' prior preprint; central to the divergence theory as condition (2.5) in Lemma 2.4. The present paper does not reprove it.
  • domain assumption s-regularity of missing-digit Cantor-Lebesgue measures, citing [31, Theorem 3.9]
    Used in sections 5.3, 5.4, and 5.6 to convert mu-measure bounds into counting bounds via moment estimates mu(B_r(y)) ~ r^s.
  • domain assumption Fourier l1 dimension estimates for special missing-digit digit sets, citing [58, Theorem A.1]
    Used in section A.1 and in section 5.5 to construct examples satisfying the dimension thresholds (1.1) and (1.3). Cited from a preprint by the co-author.

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Pith. "Pith review of Simultaneous and multiplicative Diophantine approximation on missing-digit fractals." pith.science (2026). https://pith.science/paper/67SZEHHY

@misc{pith2026241212070,
  author       = {Pith},
  title        = {Pith review of: Simultaneous and multiplicative Diophantine approximation on missing-digit fractals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/67SZEHHY}},
  note         = {Machine review of arXiv:2412.12070}
}
read the original abstract

We investigate the metric theory of Diophantine approximation on missing-digit fractals. In particular, we establish analogues of Khinchin's theorem and Gallagher's theorem, as well as inhomogeneous generalisations.

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