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REVIEW 3 major objections 4 minor 31 references

Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Expanding integer matrices, applied to almost every point of an M0-set, hit shrinking boxes with the expected number of times, up to an explicit error.

desk verdict New matrix-setting and two useful tools, but Theorem 1.8's claimed error exponent is not supported by the paper's own estimates; the Ω3 term in Lemma 3.9 forces a weaker exponent. read the letter →

arxiv 2504.21555 v1 pith:IS3TGQG6 submitted 2025-04-30 math.NT

Bo Tan , Qing-Long Zhou This is my paper · ORCID
classification math.NT MSC 28A8011J83
keywords equidistributionDiophantineapproximationM0-setsRajchmanmeasuresFourierdecayshrinkingtargetproblemexpandingintegermatricesnormalvectors
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 proves a quantitative counting theorem for Diophantine approximation on M0-sets—sets supporting a non-atomic probability measure whose Fourier transform decays at infinity—in arbitrary dimension, with denominators supplied by a sequence of expanding integer matrices. If the set $E\subset[0,1)^d$ supports a probability measure $\mu$ with Fourier decay $|\widehat{\mu}(t)|=O((\log|t|_\infty)^{-s})$ for some $s>d+1$, and the ratio matrices $A_{n+1}A_n^{-1}$ all have smallest singular value at least $K>1$, then for $\mu$-almost every $x\in E$ the number of visits of $A_n x$ to a prescribed shrinking box is $\Psi(N)$ up to an error $O(\Psi(N)^{d/(d+1)}(\log\Psi(N)+2)^{2+\varepsilon})$, where $\Psi(N)$ is the cumulative box volume. This extends the one-dimensional lacunary prototype [23] to genuinely higher-dimensional, matrix-driven dynamics. The same machinery yields quantitative equidistribution of the orbit $(A_n x)$, normality of vectors on fractals, and a zero-one law for shrinking-target intersections with $E$.

What carries the argument

The load-bearing mechanism is the A-divisibility dichotomy of Lemma 3.2: for $P=\{Ax:x\in[0,1)^d\}$, one has $\sum_{p\in P}e(-\langle k,A^{-1}p\rangle)=|\det A|$ if $(A^T)^{-1}k\in\mathbb{Z}^d$ and $0$ otherwise. This is the lattice substitute for the periodicity of arcs in the scalar proof; it makes the Fourier coefficients of the upper and lower step-function approximations $g_n^\pm$ vanish unless the dual frequency lies on the lattice $A_n^T\mathbb{Z}^d$. The remaining work is a frequency-space decomposition (Lemma 3.8) that splits correlations into three regimes, using the separation $\sigma(A_nA_m^{-1})\ge K^{n-m}$ and a $\sigma$-discrete lattice-sum estimate (Lemma 3.7) to control the overlap terms, after which the variance lemma (Lemma 3.3) converts the second-moment estimate into the almost-everywhere counting theorem.

What would settle it

Take $A_1=\begin{pmatrix}2&100\\0&2\end{pmatrix}$ and $A_n=3^{n-1}A_1$, so every ratio $A_{n+1}A_n^{-1}=3I$ has smallest singular value $3$ while $\sigma(A_1)\approx0.04<1$; for the unit vector $u$ with $\|A_1^T u\|_2\approx0.04$, the key lower bound $\|(A_2-A_1)^T u\|_2\ge K^{0}(K^1-1)=2$ fails because the true value is about $0.08$. Checking whether the counting estimate of Theorem 1.8 still holds for this sequence separates a harmless missing constant from a genuinely false statement.

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Extended reading notes

Core claim

The central claim is that the one-dimensional q-divisibility of the classical proof can be replaced by an A-divisibility principle on $\mathbb{Z}^d$: for an integral matrix $A$, the exponential sum over the complete residue set $P=\{Ax:x\in[0,1)^d\}$ equals $|\det A|$ precisely when $(A^T)^{-1}k\in\mathbb{Z}^d$, and vanishes otherwise. With this principle, the paper pushes the step-function and variance-method estimates through for arbitrary expanding integer matrices, not only diagonal ones. The main theorem (Theorem 1.8) then asserts that, under polylogarithmic Fourier decay with $s>d+1$ and the uniform ratio bound $\sigma(A_{n+1}A_n^{-1})\ge K>1$, the hitting count satisfies $R(x,N)=\Psi(N)+O(\Psi(N)^{d/(d+1)}(\log\Psi(N)+2)^{2+\varepsilon})$ for $\mu$-almost every $x$, where $\Psi(N)=\sum_{n=1}^N 2^d r_1(n)\cdots r_d(n)$. From this the paper derives equidistribution of $(A_n x)$, Schmidt normality of vectors on fractals, and a Borel-Cantelli dichotomy for the shrinking target set.

Load-bearing premise

The proof assumes that every matrix $A_m$ already expands all directions by at least $K^{m-1}$, which forces $\sigma(A_1)\ge 1$; the stated hypotheses (expanding eigenvalues and $\sigma(A_{n+1}A_n^{-1})\ge K>1$) do not guarantee this, so the displayed lower bound $K^{m-1}(K^{n-m}-1)$ is missing a factor of $\sigma(A_1)$.

Editorial extensions

If this is right

  • For $\mu$-almost every $x\in E$, the sequence $(A_n x)$ modulo 1 is equidistributed whenever $\widehat{\mu}$ has polylogarithmic decay with $s>d+1$ and the ratio matrices expand by at least $K>1$.
  • Taking all $r_i(n)\equiv a_i$ constant recovers the expected hitting frequency $2^d a_1\cdots a_d$ with an explicit error term, making the qualitative equidistribution statement quantitative.
  • The shrinking target set $W_A(y,(R_n))\cap E$ has $\mu$-measure 0 when $\sum r_1(n)\cdots r_d(n)<\infty$ and 1 when the sum diverges: a zero-one law for matrix transformations on fractal measures.
  • For diagonal matrices such as $A_n=\operatorname{diag}(2^n,3^n)$ the coordinates decouple and the problem reduces to one-dimensional Fourier estimates, while non-diagonal matrices remain genuinely higher-dimensional; both are covered by the theorem.
  • As a consequence, every measure with sufficiently fast Fourier decay on a fractal carries a set of full measure whose points are normal with respect to the given expanding matrix sequence.

Reading between the lines

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

  • Setting $d=1$ in the theorem gives an error term $\Psi(N)^{1/2}(\log\Psi(N))^{2+\varepsilon}$, weaker than the $\Psi(N)^{2/3}$ error of the scalar prototype; the authors' own remark expects the exponent to be suboptimal, and a sharper lattice-sum estimate would likely improve it uniformly in $d$.
  • Because the transference only uses the overlattice structure $A^{-1}\mathbb{Z}^d$ and the ratio separation $\sigma(A_{n+1}A_n^{-1})\ge K$, the same argument should extend to rational or non-integer expanding matrices and to toral endomorphisms of any modulus, giving quantitative shrinking-target statements beyond integral matrices.
  • A natural test of the singular-value hypothesis is to study sequences whose first matrix is an expanding shear with $\sigma(A_1)<1$; if the expected-count formula still holds, the theorem's hypotheses could be relaxed to an explicit dependence on $\sigma(A_1)$ rather than requiring it to be at least 1.
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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 / 4 minor

Summary. This paper studies quantitative Diophantine approximation on M0-sets in R^d when the approximating denominators are drawn from a sequence of expanding integral matrices. The main result, Theorem 1.8, asserts that for a probability measure mu supported on E with polylogarithmic Fourier decay |hat{mu}(t)| = O((log|t|_infty)^{-s}) for s>d+1, and for a matrix sequence satisfying sigma(A_{n+1}A_n^{-1}) >= K > 1, the hitting count of the orbit (A_n x) in shrinking rectangles satisfies R(x,N) = Psi(N) + O(Psi(N)^{d/(d+1)} (log Psi(N)+2)^{2+epsilon}) for mu-almost all x. The paper also proves an equidistribution theorem (Theorem 1.7) and a shrinking-target zero-full law (Theorem 1.10). The technical machinery includes an A-divisibility lemma for exponential sums over dual lattices and a correlation estimate based on the singular value decomposition.

Significance. The framework is attractive: it extends the one-dimensional PVZZ counting theorem to a genuinely higher-dimensional matrix setting, and the applications to normality of vectors and shrinking targets on fractals are relevant. The proof idea of using A-divisibility and singular-value separation to control overlapping measures is a natural and potentially useful contribution. However, the headline quantitative error term in Theorem 1.8 is not supported by the manuscript's own estimates; the proof of the key correlation lemma contains an exponent-balancing error. As a result, the advertised improvement over the scalar case is unproved, although a weaker error term may still yield a new theorem for d>=2.

major comments (3)
  1. [Section 3.4, Lemma 3.9] The conclusion of Lemma 3.9 does not follow from the displayed estimates. In Case 2 of the proof, the Omega_3 contribution from Lemma 3.8 is bounded by sum_{a<=m<n<=b} epsilon(m)^{-d/2} psi(n) K^{-(n-m)} and then by (sum psi)^{1+d*delta/2} via geometric series control. Setting delta=2/(d+1), as the text does, gives 1+d*delta/2 = (2d+1)/(d+1), which is larger than the claimed exponent 2d/(d+1). The final display in Lemma 3.9 nevertheless caps this term at Psi^{2d/(d+1)}, which is internally inconsistent. Optimizing the two competing error terms 2-delta and 1+d*delta/2 yields delta=2/(d+2) and a variance exponent 2(d+1)/(d+2); consequently Lemma 3.3 gives an error exponent (d+1)/(d+2), not d/(d+1). For d=1 this recovers the scalar PVZZ exponent 2/3 rather than the claimed 1/2. Theorem 1.8 must either be restated with the weaker exponent or the Omega_3 sum needs a substantially improved estimate.
  2. [Section 3.4, Lemma 3.9] The displayed upper-bound decomposition in Lemma 3.9 contains the term sum_{a<=m<n<=b} n^{-s} epsilon(m)^{-d/2} epsilon(n)^{-d/2}, which comes from Case Omega_1 of Lemma 3.8. This term is never bounded in the subsequent analysis of either Case 1 or Case 2. Although it can probably be controlled using (3.10) and s>d+1, the estimate is absent from the proof and needs to be supplied for the correlation bound to be complete.
  3. [Section 2, proof of Theorem 1.7; Section 3.4, Lemma 3.8] The proof of Theorem 1.7 and the Case 1 and Case 2 estimates of Lemma 3.8 use the inequality sigma(A_m) >= K^{m-1} without stating the necessary constant. This requires sigma(A_1) >= 1, which is not implied by the hypotheses that A_n are expanding integral matrices and sigma(A_{n+1}A_n^{-1}) >= K > 1; for example, a matrix with large off-diagonal entries can have eigenvalues larger than 1 while having a smallest singular value smaller than 1. The displayed bound ||(A_n-A_m)^T k||_2 >= K^{m-1}(K^{n-m}-1) is therefore false as written. The proofs should carry the constant sigma(A_1) > 0 and choose m0 (or adjust the implicit constants) depending on it; this is likely a minor fix, but it is needed for correctness.
minor comments (4)
  1. [Section 2, proof of Theorem 1.7] The manuscript contains a garbled inequality in the line that chooses m0: it reads 'Km−1≥ 1 + 1 K− 1≥ 1 + 1 Kn−m− 1', which appears to be missing superscripts and is not readable as a mathematical statement. Please rewrite this line correctly.
  2. [Section 3.3, Lemma 3.5] In Lemma 3.5, the definition of epsilon(n) uses S_n = sum_{k=1}^n psi(k), while later in Section 3.4 the same symbol epsilon(n) is redefined with sums starting at a. This potential for confusion should be addressed by using distinct notation or by explicitly noting the local redefinition.
  3. [Section 3.3, proof of Lemma 3.5] The estimate n^{-s} epsilon(n)^{-d/2} << n^{-s+d/2} implicitly ignores the power 1+xi in the definition epsilon(n) = min{1, S_n^{-1-xi}}. For the series to converge, xi must be chosen small enough; the text says 's>1+d' but the calculation should show explicitly that d(1+xi)/2 < s-1, which is possible since s>d+1 and xi can be taken sufficiently small.
  4. [Introduction, Section 1.2] The phrase 'expanding integral matrices' is used but not formally defined. Since the proof relies on the singular value condition sigma(A_{n+1}A_n^{-1}) >= K > 1, it would be helpful to state explicitly that each A_n is assumed nonsingular and to clarify the relationship between 'expanding' and the singular value hypothesis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.8 is derived from explicit Fourier-decay and singular-value hypotheses using independent lemmas.

full rationale

The derivation chain is self-contained: the measure mu with polylogarithmic Fourier decay and the matrix sequence with uniform lower bounds on minimal singular values of An+1A_n^{-1} are hypotheses, not fitted parameters; the proof proceeds by explicit Fourier estimates (Lemma 3.4), lattice sparsity (Lemma 3.7), and a standard variance/counting lemma (Lemma 3.3). Prior work [23] and the authors' own [30] are cited as background and context, not as load-bearing inputs, and no parameter is tuned to reproduce the target result. The claimed error exponent is obtained by the paper's own estimates plus the explicit choice delta = 2/(d+1); whether that balancing is quantitatively correct is a mathematical correctness issue, not circularity. Likewise, any gap about the initial singular value sigma(A_1) or the S3/Omega3 term concerns validity of the proof, not reduction of the conclusion to its inputs. No self-definitional, fitted-input-called-prediction, self-citation-chain, or renaming pattern is present.

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

The central claim rests on the Fourier decay and singular value hypotheses plus standard Fourier-analytic tools. No new entities are postulated and no constants are fitted to data.

assumptions (5)
  • domain assumption Fourier decay hypothesis |\hat{μ}(t)| ≪ (log|t|∞)^{-s} with s>d+1
    This is the main analytic input; it is used at several points (Lemma 3.5, Lemma 3.8) to control the expectation and correlation sums. The paper assumes it, does not prove it for any specific fractal.
  • domain assumption Minimal singular value condition σ(A_{n+1}A_n^{-1}) ≥ K>1 for all n
    This is the dynamical hypothesis defining the matrix sequence; it drives the lower bounds on ||A_n^T k|| and the sparsity of the lattice Γ in Lemma 3.7.
  • ad hoc to paper Unstated premise σ(A_1) ≥ 1 used to get σ(A_m) ≥ K^{m-1}
    Not in the theorem statement; used in the proofs of Theorem 1.7 and Lemma 3.8. Repair: carry the constant σ(A_1) and shift m0.
  • domain assumption Polynomial lower bound r_i(n) ≥ n^{-τ} assumed without loss of generality
    Section 3.4, after (3.6). The proof replaces r_i by max{r_i, n^{-τ}} and argues the counting functions differ by O(1) μ-a.e., using convergence Borel-Cantelli. This is an additional regularity condition on the shrinking radii.
  • standard math Standard tools: Weyl criterion, Davenport-Erdős-LeVeque lemma (Lemma 2.1), Philipp's quantitative Borel-Cantelli (Lemma 3.3), Poisson summation duality for lattices (Lemma 3.1)
    Unproved background results cited from [6], [8], [22] and classical Fourier analysis.

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Pith. "Pith review of Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets." pith.science (2026). https://pith.science/paper/IS3TGQG6

@misc{pith2026250421555,
  author       = {Pith},
  title        = {Pith review of: Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IS3TGQG6}},
  note         = {Machine review of arXiv:2504.21555}
}
abstract

Let $E\subset [0,1)^{d}$ be a set supporting a probability measure $\mu$ with Fourier decay $|\widehat{\mu}({\bf{t}})|\ll (\log |{\bf{t}}|)^{-s}$ for some constant $s>d+1.$ Consider a sequence of expanding integral matrices $\mathcal{A}=(A_n)_{n\in\N}$ such that the minimal singular values of $A_{n+1}A_{n}^{-1}$ are uniformly bounded below by $K>1$. We prove a quantitative Schmidt-type counting theorem under the following constraints: (1) the points of interest are restricted to $E$; (2) the denominators of the ``shifted'' rational approximations are drawn exclusively from $\mathcal{A}$. Our result extends the work of Pollington, Velani, Zafeiropoulos, and Zorin (2022) to the matrix setting, advancing the study of Diophantine approximation on fractals. Moreover, it strengthens the equidistribution property of the sequence $(A_n{\bf x})_{n\in\N}$ for $\mu$-almost every ${\bf x}\in E.$ Applications include the normality of vectors and shrinking target problems on fractal sets.

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