Exact Diophantine approximation$\colon$ the simultaneous case in $\mathbb{R}^{2}$

Bo Tan; Qing-Long Zhou

REVIEW 1 major objections 4 minor 17 references

Exact Diophantine approximation$\colon$ the simultaneous case in $\mathbb{R}^{2}$

T0 review · 1 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that the set of pairs in the plane with exact simultaneous approximation order ψ has Hausdorff dimension 3/(λ+1), the same as the full well-approximable set.

desk verdict Elementary proof of a known dimension formula; the printed divergence estimate has a fixable inequality slip, but the argument's core is sound. read the letter →

arxiv 2411.18439 v1 pith:GUKNKTNN submitted 2024-11-27 math.NT

Bo Tan , Qing-Long Zhou This is my paper · ORCID

classification math.NT MSC 28A8011K5511J83

keywords HausdorffdimensionexactapproximationordersimultaneousDiophantineMassTransferencePrincipleMarstrandslicinglemmafiberwell-approximableset

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 that in the plane, the set of pairs whose simultaneous Diophantine approximation by rationals has exactly the prescribed error order $\psi$ has Hausdorff dimension $3/(\lambda+1)$, where $\lambda$ is the growth exponent of $\psi$. This is the same dimension as the well-approximable set $W(2,\psi)$, so requiring the approximation to be exact rather than merely at least as good does not shrink the fractal size. The result fills the last unproved case $n=2$ in the family of dimension formulas for Exact($n,\psi$), after the one-dimensional case and the cases $n\ge 3$ were known. The proof splits into a lower bound using Marstrand's slicing lemma together with a mass transference argument, and an upper bound inherited from the dimension formula for $W(2,\psi)$.

What carries the argument

The proof rests on two tools. Marstrand's Slicing Lemma says that if a set $E\subset X\times Y$ has sections of dimension at least $t$ over a base of dimension $s$, then $\dim_H E\ge s+t$. The Mass Transference Principle says that for a dimension function $f$ with $f(x)/x$ monotonic, if the series $\sum_q f(\psi(q)/q)\varphi(q)$ diverges, then the Hausdorff $f$-measure of the set of numbers with a coprime approximation of quality $\psi$ is full. The paper's twist is to feed the principle the auxiliary function $\Psi(q)=\psi(q)$ for $q\in Q(x_1)$ and $0$ otherwise, where $Q(x_1)$ is the set of denominators for which $x_1$ is $\psi$-approximable; this proves the fiber lower bound. The divergence of the relevant series then follows from the definition of $\lambda$ and the standard estimate for products over primes.

What would settle it

Take $\psi(q)=q^{-\lambda}$ for some $\lambda>1$ and an explicit $x_1\in\mathrm{Exact}(1,\psi)$ whose approximating denominator set $Q(x_1)$ is known, then evaluate the series $\sum_{q\in Q(x_1)}(\psi(q)/q)^s\varphi(q)$ at $s=(1-\varepsilon)/(\lambda+1)$. A convergent value for any such series would break the proof's divergence claim; a direct computation of $\dim_H \mathrm{Exact}(2,\psi)$ differing from $3/(\lambda+1)$ for any $\lambda$ would disprove the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: if $q\psi(q)$ is non-increasing and tends to zero at infinity, then $\dim_H \mathrm{Exact}(2,\psi)=3/(\lambda+1)$, where $\lambda := \lim_{q\to\infty} -\log \psi(q)/\log q$. The discovery is that exactness does not reduce the dimension in two dimensions: for every $x_1\in \mathrm{Exact}(1,\psi)$, the fiber of $x_2$ for which $(x_1,x_2)$ has exact order $\psi$ has dimension at least $1/(\lambda+1)$, and Marstrand's slicing lemma converts this fiber lower bound into the full lower bound. The upper bound is immediate from the inclusion $\mathrm{Exact}(2,\psi)\subset W(2,\psi)$ and the known dimension of $W(2,\psi)$. The argument works by thinning $\psi$ to an auxiliary function $\Psi$ supported only on the denominators $q$ for which $qx_1$ is already $\psi$-close, so that the mass transference principle can be applied to a subset of the fiber.

Load-bearing premise

The lower-bound argument applies the Mass Transference Principle to the auxiliary function $\Psi$, which is zero except on the possibly very sparse set $Q(x_1)$ of denominators that approximate $x_1$. The paper does not spell out that this sparse, non-monotone function meets the principle's hypotheses, and the displayed divergence estimate in Proposition 2.3 contains a reversed inequality.

Editorial extensions

If this is right

For $n=2$, the set of exactly approximable pairs has the same Hausdorff dimension as the full well-approximable set $W(2,\psi)$.
Combined with the known cases $n=1$ and $n\ge 3$, the formula $\dim_H\mathrm{Exact}(n,\psi)=(n+1)/(\lambda+1)$ now holds for every dimension $n\ge 1$ under the same monotonicity assumption.
For every $x_1\in\mathrm{Exact}(1,\psi)$, the fiber of $x_2$ with $(x_1,x_2)\in\mathrm{Exact}(2,\psi)$ has Hausdorff dimension at least $1/(\lambda+1)$.
The upper bound requires no new work: it follows from the inclusion $\mathrm{Exact}(2,\psi)\subset W(2,\psi)$ and the known dimension of $W(2,\psi)$.

Reading between the lines

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

If the sparse-support trick can be made fully rigorous, the same induction through fibers could establish the exact-dimension formula for all $n$ without relying on parametric geometry of numbers.
The equality of dimensions suggests a general principle: in simultaneous Diophantine approximation, exactness is dimension-neutral, deleting the faster-than-$\psi$ approximation layers without changing the fractal exponent.
A testable extension is to allow coordinate-dependent error functions $\psi_1,\psi_2$; the natural conjecture is that $\dim_H\mathrm{Exact}(2,(\psi_1,\psi_2))$ is governed by the slower error function.

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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: the paper proves Theorem 1.1, but the statement is already in Bandi–De Saxcé [1], which the authors cite. The novelty is the proof, not the result. The proof is short and genuinely simpler than the parametric-geometry route: fiber slicing plus the one-dimensional exact-order result plus MTP. For the n=2 case this is a clean, readable argument.

The proof as printed has a real slip in Proposition 2.3. The chain (ψ(q)/q)^s φ(q) ≥ q^{-(1−ε)} φ(q) does not follow from the definition of λ. Since λ = lim −log ψ/log q, for large q we have ψ(q) ≥ q^{-(λ+ε)} and ψ(q) ≤ q^{-(λ−ε)}. To lower-bound (ψ/q)^s you need the smallest ψ, which gives (ψ/q)^s ≥ q^{-s(λ+1+ε)} = q^{-(1−ε)−sε}. That exponent is less than −(1−ε), not greater, so the displayed inequality is reversed. The good news: the corrected bound still diverges. The series becomes ∑_{q∈Q(x1)} q^{-(1−ε)−sε} φ(q) ≫ ∑ q^{ε−sε}/log q = ∑ q^{ε(1−s)}/log q, which diverges for any infinite Q(x1) because s<1. So the MTP conclusion survives after replacing the wrong exponent. The error is fixable and the theorem stands, but the written argument is not sound as it stands.

A smaller point: the MTP lemma is cited without checking that the sparse, non-monotone Ψ satisfies its hypotheses. In practice MTP does not require monotonicity of the approximating function, so this is more an exposition gap than a mathematical hole, but it should be addressed.

The paper is honest: it cites [1] and explicitly says the proof is different. The abstract's "fill a gap" refers to Fregoli's approach, which is fair. The citation pattern is clean, no self-citation issue.

Who is this for? Anyone working on exact approximation or wanting an elementary proof of the n=2 dimension formula. The paper does not change the landscape, but it is a useful pedagogical and technical contribution. I would send it to a referee if the authors fix the inequality and add a remark that the result is already known; the referee should check the corrected exponent.

Recommendation: engage with it after revision.

Referee Report

1 major / 4 minor

Summary. The paper proves that for a function ψ satisfying that qψ(q) is non-increasing and tends to 0 at infinity, with λ = lim_{q→∞} −log ψ(q)/log q, the Hausdorff dimension of the set Exact(2,ψ) of vectors in R² having exact simultaneous approximation order ψ equals 3/(λ+1). The proof combines the one-dimensional exact-order result of Bugeaud and Bugeaud–Moreira with a fibre inclusion inspired by Fregoli, Marstrands' slicing lemma, and the mass transference principle for Duffin–Schaeffer sets. The upper bound is obtained from Rynne's formula for the simultaneous well-approximable set W(2,ψ).

Significance. If correct, the paper fills the n=2 gap in Fregoli's fibre-based method and provides an elementary proof of the exact-order dimension for the simultaneous case in R². The result itself is not entirely new—Bandi–De Saxcé have treated all n via parametric geometry—but the proof here is short and uses different, standard tools. The main ingredients are cited correctly and the overall strategy is natural. However, there is a local but load-bearing error in the divergence estimate in Proposition 2.3, which must be repaired before the proof is sound.

major comments (1)

[§2, Proposition 2.3] The displayed chain of inequalities in the proof of Proposition 2.3 contains a reversed inequality. Since λ = lim_{q→∞} (−log ψ(q))/log q, for every ε>0 and all sufficiently large q one has q^{−(λ+ε)} ≤ ψ(q) ≤ q^{−(λ−ε)}. Therefore (ψ(q)/q)^s ≥ q^{−s(λ+1+ε)} = q^{−(1−ε)−sε}, which is strictly smaller than the claimed bound q^{−(1−ε)} for q>1. Thus the inequality (ψ(q)/q)^s ≥ q^{−(1−ε)} is false. This is load-bearing because the divergence of ∑_{q∈Q(x1)} (ψ(q)/q)^s φ(q) is what triggers the mass transference principle. The error is local and repairable: replacing q^{−(1−ε)} with q^{−(1−ε)−sε} yields the lower bound ∑_{q∈Q(x1)} q^{ε(1−s)}/log q, which diverges because Q(x1) is infinite and ε(1−s)>0. The proposition's conclusion then follows, but the proof as printed is not sound.

minor comments (4)

[Introduction, definition of W*(1,ψ)] The definition of W*(1,ψ) contains a stray symbol γ: '||qx − γ||_*' should read '||qx||_*'.
[Lemma 2.2] In the statement of Lemma 2.2 the summation is written as ∑_{q∈N} f(ψ(n)/n) φ(n), mixing the variables q and n; it should be ∑_{q∈N} f(ψ(q)/q) φ(q). Also, since the lemma imposes monotonicity of x^{-1}f(x), the intended direction (non-increasing or non-decreasing) should be specified, as it is relevant to the application f(x)=x^s.
[§2, Proposition 2.3] The inclusion W*(1,Ψ) ⊂ W(x1,1,ψ) is only stated as 'readily checked'. A short verification would improve readability: for any c<1, the exactness of x1 implies that all but finitely many q satisfy ||qx1|| ≥ cψ(q), so points in W*(1,Ψ) cannot lie in W(2,cψ) and therefore lie in Exact(2,ψ).
[§1, upper bound] The upper bound is derived from Rynne's formula (1.1). If Rynne's theorem is stated under a different monotonicity assumption (for instance, ψ decreasing rather than qψ(q) non-increasing), the exact hypothesis should be cited. Alternatively, the needed inequality dim_H Exact(2,ψ) ≤ 3/(λ+1) follows from the standard covering construction for W(2,ψ) and does not require monotonicity beyond the asymptotic of ψ and ψ(q) ≤ 1/(2q).

Circularity Check

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No circularity: the dimension result is derived from independent external theorems (Bugeaud's exact-order result, Rynne's formula, Marstrand's slicing lemma, and the Beresnevich–Velani mass transference principle) and no step assumes the target formula. The only flagged issue is a reversed inequality in Proposition 2.3, which is a soundness gap in the proof as printed, not a circular reduction.

full rationale

The paper's derivation chain is self-contained in the circularity sense. The upper bound dim_H Exact(2,ψ) ≤ 3/(λ+1) is obtained by monotonically covering by Rynne's known dimension formula (1.1) for W(2,ψ); no fitted coefficient or assumed conclusion is involved. The lower bound combines Marstrand's Slicing Lemma with the one-dimensional exact-order theorem dim_H Exact(1,ψ)=2/(λ+1) of Bugeaud and Bugeaud–Moreira, cited as an external result. Proposition 2.3 then proves the remaining slice dimension lower bound by constructing Ψ(q)=ψ(q) for q∈Q(x1) and 0 otherwise, and by applying the Mass Transference Principle of Beresnevich–Velani; W*(1,Ψ)⊆W(x1,1,ψ) is a genuine set inclusion relying on x1∈Exact(1,ψ), not on the target dimension. There are no fitted parameters, no ansatz smuggled in by self-citation (the bibliography contains no prior work of the present authors), and no theorem is imported from the authors' own earlier papers. The only substantive defect visible in the text is mathematical rather than circular: in Proposition 2.3 the displayed estimate (ψ(q)/q)^s ≥ q^{-(1-ε)} uses the wrong side of the definition of λ; from ψ(q) ≥ q^{-(λ+ε)} one obtains only (ψ(q)/q)^s ≥ q^{-(1-ε)-sε}, which is weaker than q^{-(1-ε)}. That weakened bound still makes the series diverge (terms q^{ε(1-s)}/log q), so the error appears repairable and does not amount to assuming the theorem.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no free parameters and no new entities. Its central claim rests on external theorems: Marstrand slicing, the mass transference principle, Rynne's formula, and the one-dimensional exact order result, plus Mertens' theorem. The most fragile input is the applicability of the MTP to the sparse auxiliary function Ψ, which the authors assert without checking the hypotheses.

assumptions (7)

standard math Marstrand's slicing lemma (Lemma 2.1) as stated in Falconer [8]
Used to convert fiber dimension estimates into a lower bound for the product-like set Exact(2,ψ).
standard math Mass Transference Principle of Beresnevich and Velani (Lemma 2.2)
Used to show dim W*(1,Ψ) ≥ s from divergence of ∑ (Ψ(q)/q)^s φ(q).
domain assumption The MTP applies to the sparse, non-monotone function Ψ supported on Q(x1)
The authors apply the lemma to Ψ without explicitly checking that the theorem's hypotheses are satisfied for this non-monotone, potentially sparse function.
standard math Rynne's formula dim_H W(n,ψ) = (n+1)/(λ+1), equation (1.1)
Gives the upper bound for Exact(2,ψ) via containment in W(2,ψ).
standard math dim_H Exact(1,ψ) = 2/(λ+1), equation (1.2)
Base dimension used in Marstrand slicing for the lower bound.
standard math Mertens' theorem: φ(q)/q ≫ 1/log q
Used to convert the number-theoretic sum into a divergent series.
domain assumption qψ(q) is non-increasing and tends to zero at infinity
This is the standing hypothesis of Theorem 1.1 and of the cited exact approximation results (1.2) and (1.1).

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Cite this review

Pith. "Pith review of Exact Diophantine approximation$\colon$ the simultaneous case in $\mathbb{R}^{2}$." pith.science (2026). https://pith.science/paper/GUKNKTNN

@misc{pith2026241118439,
  author       = {Pith},
  title        = {Pith review of: Exact Diophantine approximation$\colon$ the simultaneous case in $\mathbbR^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GUKNKTNN}},
  note         = {Machine review of arXiv:2411.18439}
}

read the original abstract

We fill a gap in the study of the Hausdorff dimension of the set of exact approximation order considered by Fregoli [Proc. Amer. Math. Soc. 152 (2024), no. 8, 3177--3182].

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Bandi and N

P. Bandi and N. De Saxc´ e. Hausdorﬀ dimension and exact appro ximation order in Rn. arXiv : 2312.10255

arXiv

[2] [2]

Beresnevich and S

V. Beresnevich and S. Velani. A mass transference principle and the Duﬃn-Schaeﬀer conjec ture for Hausdorﬀ measures. Ann. of Math. (2) 164 (2006), no. 3, 971–992

work page 2006

[3] [3]

Y. Bugeaud. Sets of exact approximation order by rational numbers. Math. Ann. 327 (2003), no. 1, 171–190

work page 2003

[4] [4]

Y. Bugeaud. Sets of exact approximation order by rational numbers. II. Unif. Distrib. Theory 3 (2008), no. 2, 9–20

work page 2008

[5] [5]

Bugeaud and C

Y. Bugeaud and C. Moreira. Sets of exact approximation order by rational numbers III. Acta Arith. 146 (2011), no. 2, 177–193

work page 2011

[6] [6]

T. Das, L. Fishman, D. Simmons and M. Urba´ nski. A variational pr inciple in the parametric geometry of numbers. Adv. Math. 437 (2024), Paper No. 109435 , 130 pp

work page 2024

[7] [7]

Duﬃn and A.C

R.J. Duﬃn and A.C. Schaeﬀer. Khintchine’s problem in metric Diophantine approximation . Duke Math. J. 8 (1941), 243–255

work page 1941

[8] [8]

Falconer, Fractal geometry

K.J. Falconer, Fractal geometry. Mathematical foundations and applicati ons. Third edition. John Wiley & Sons, Ltd., Chichester, 2014

work page 2014