A note on limsup sets of annuli

Benjamin Ward; Mumtaz Hussain

REVIEW 3 major objections 3 minor 20 references

A note on limsup sets of annuli

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper derives an exact Hausdorff dimension formula for the set of points that fall inside infinitely many max-norm annuli centered at rational points, and extends it to norm-difference and rectangular annuli.

desk verdict Genuine new result in Theorem 1.1, but two advertised theorems have load-bearing proof gaps; repairable, but not acceptable as is. read the letter →

arxiv 2502.09807 v1 pith:Z2UHYUMI submitted 2025-02-13 math.NT math.DSmath.MG

Mumtaz Hussain , Benjamin Ward This is my paper · ORCID

classification math.NTmath.DSmath.MG MSC 11J8328A8011K60

keywords HausdorffdimensionDiophantineapproximationlimsupsetsannulimasstransferenceprincipleJarník–Besicovitchtheoremexactorder

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 studies the set of points in $\mathbb{R}^n$ that lie in infinitely many annuli centered at rational points, where each annulus is the difference between a max-norm ball of radius $\psi(q)/q$ and an inner ball of radius $(1-\varphi(q))\psi(q)/q$. The main result is an exact Hausdorff dimension formula for this limsup set: for $\psi(q)=q^{-\tau_\psi}$ and $\varphi(q)=q^{-\tau_\varphi}$ with $\tau_\psi\ge 1/n$, the dimension equals $\min\{(n+1)/(1+\tau_\psi),\ (n+1+(n-1)\tau_\varphi)/(1+\tau_\psi+\tau_\varphi)\}$. A curious consequence is that in dimensions $n\ge 2$, if the outer radii shrink slowly enough and the annuli become very thin, the dimension tends to $n-1$, while if the outer radii shrink fast, the inner thickness is irrelevant. The paper also proves analogues for annuli defined by two different norms and for rectangular annuli, and relates the results to recent work on exact approximation order.

What carries the argument

The load-bearing machinery is a shifted mass transference principle: Theorem 3.2 for balls and Theorem 3.4 for hyperrectangles. These combine Cassel's scaling lemma (in the cleaned-up form of Lemma 3.1) with the Beresnevich–Velani mass transference principle and the Wang–Wu mass transference principle from rectangles to rectangles. The shifting step is what makes annuli tractable: a thin annulus is not a ball, but it is designed to contain a ball or box of comparable size whose centre is moved slightly away from the rational point, and the shifted mass transference principle says such centre moves are harmless provided they are small compared with the blown-up radii used in the divergence condition. For rectangular annuli, the annulus is decomposed into $2n$ hyperrectangles, each missing only one coordinate side, and the rectangle mass transference principle supplies the lower-bound dimension.

What would settle it

Check the containment claimed in the proof of Theorem 1.2 with $n=2$, $\rho=2$, and outer radius $r=1$: the proposed shifted ball reaches max-norm distance about $1.77$ from the rational centre, exceeding the outer radius $1$, so it is not contained in the annulus.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.1: the Hausdorff dimension of $W_n(\psi,\varphi)$ is exactly $\min\{(n+1)/(1+\tau_\psi),\ (n+1+(n-1)\tau_\varphi)/(1+\tau_\psi+\tau_\varphi)\}$ for power-decay rates with $\tau_\psi\ge 1/n$. When $\tau_\psi\le 2/(n-1)$, the second term controls the dimension and, as the annuli become thinner ($\tau_\varphi\to\infty$), the dimension tends to $n-1$, which is strictly smaller than the classical Jarn\'ik–Besicovitch dimension unless $\tau_\psi=2/(n-1)$. When $\tau_\psi\ge 2/(n-1)$, the first term wins and the dimension is exactly the classical ball-approximation value, so the inner boundary of the annulus is irrelevant. The paper further claims Theorem 1.2, that annuli formed as the difference of max-norm and $\rho$-norm balls have the same dimension $(n+1)/(1+\tau_\psi)$ for any $\rho$, and Theorem 1.3, a general max-min formula for rectangular annuli with coordinate-dependent rates, from which Theorem 1.1 follows as the symmetric case.

Load-bearing premise

The lower-bound dimension results rest on containment estimates that place a full, slightly shifted ball or box inside each annulus with controlled distortion; if those estimates fail for some choice of norms or parameters, the corresponding lower bound is unsupported.

Editorial extensions

If this is right

If Theorem 1.1 is correct, then for $n\ge 2$ and $\tau_\psi<2/(n-1)$ the Hausdorff dimension of $W_n(\psi,\varphi)$ collapses to $n-1$ in the limit of negligibly thin annuli, even though the approximating rational points remain dense.
For $\tau_\psi\ge 2/(n-1)$ the dimension is exactly the classical Jarn\'ik–Besicovitch value $(n+1)/(1+\tau_\psi)$, so thinning the annuli loses nothing in dimension.
For exact approximation order, Theorem 1.1 gives upper bounds on $\dim_H \mathrm{Exact}_n(f,\psi)$ that are strictly smaller than $\dim_H \mathrm{Exact}_n(\psi)$ when $\tau_\psi<2/(n-1)$ and $f(q)=1-q^{-\varepsilon}$; in the complementary range the dimensions may coincide.
Theorem 1.2 says the distinction between the max norm and any $\rho$-norm is invisible in Hausdorff dimension: the quasi-annuli have the same dimension as plain $\psi$-approximable balls.
Theorem 1.3 extends the formula to rectangular annuli with different decay rates in each coordinate, recovering Theorem 1.1 as the symmetric case.

Reading between the lines

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

The dimension-tends-to-$(n-1)$ phenomenon mirrors a known effect in weighted Diophantine approximation, but the paper notes that rational points on coordinate hyperplanes cannot explain it here; a plausible mechanism is that the limiting set concentrates near a codimension-one submanifold, which could be tested numerically by looking at the distribution of approximating points inside the thin annu
The shifted mass transference principles are likely to apply well beyond annuli: any 'shell' obtained as the difference of two balls or boxes that contains a small box in a suitable location should yield analogous dimension formulas, for example shells that are thin only in one coordinate direction.
The perturbed approximation theorem (Theorem 2.1) suggests a natural Duffin–Schaeffer analogue in which rationals $p/q$ are replaced by $p/q+\gamma(p,q)/q$ with $\gamma$ small; the paper alludes to this via the Koukoulopoulos–Maynard theorem but does not develop it, so a perturbed Duffin–Schaeffer statement is a concrete open extension.

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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

First thing to know: this is a mixed bag. The paper contains a genuine new result, Theorem 1.1, which gives the Hausdorff dimension of the limsup set of max-norm annuli centred at rationals. The formula is natural, the equal-exponent proof largely works, and the connection to exact approximation order is interesting. The shifted mass transference principles (Theorems 3.2 and 3.4) are a modest but useful extension of the standard MTP, and the perturbation theorem 2.1 is a nice tool in itself.

But the paper overreaches. Theorem 1.2's lower-bound proof relies on a geometric containment that is false. The claimed shifted ball B∞(p/q + ... , r(n^{1/ρ}-n^{-1/ρ})) is not contained in the annulus. For n=2, ρ=2, R=1, the shift is roughly 1.06 in each coordinate and the radius 0.707; points in that ball can have max norm above 1, and can also have Euclidean norm below 1. So the inclusion fails in both directions. The theorem may still be true, and the proof can likely be repaired by using a much smaller ball near the corner of the cube, but as written the lower bound is unproved.

Theorem 1.3 also has a gap. In the second lower-bound case, the proof assumes the existence of an integer ℓ satisfying τψℓ > (1 - ∑_{i>ℓ} τψi)/ℓ. When ∑τψi = 1 and some τψi < 1/n, no such ℓ may exist (e.g., n=2, τψ1+τψ2=1, τψ2<1/2). The boundary case is exactly the one that matters for the theorem's stated range ∑τψi ≥ 1. This is not a tiny slip; it leaves the lower bound incomplete for that case.

What is good: Theorem 1.1 itself seems solid, the upper bounds are standard, and the paper is honest about what it does. The flaws look repairable, and the central formula may well be correct. But as submitted, two of the three advertised theorems are not proved.

Who this is for: people working in metric Diophantine approximation and fractal geometry. A serious referee should see it, because Theorem 1.1 is a natural result and the repair path is clear. My recommendation: send it to peer review with a request for major revision, focusing on fixing the inclusion in Theorem 1.2 and handling the boundary case in Theorem 1.3.

Referee Report

3 major / 3 minor

Summary. This note studies limsup sets of annuli centred at rational points in R^n. Theorem 1.1 states an exact Hausdorff dimension formula for the set W_n(ψ,φ) of points lying in infinitely many max-norm annuli with outer radius ψ(q)/q and inner radius (1−φ(q))ψ(q)/q. Theorem 1.2 treats annuli obtained by removing a ρ-norm ball from a max-norm ball, and Theorem 1.3 states a weighted rectangular analogue. The proofs are based on a shifted mass transference principle (Theorems 2.1, 3.2, 3.4) obtained by combining Cassel's scaling lemma with the Beresnevich–Velani and Wang–Wu mass transference principles. The announced results include the phenomenon that, for n ≥ 2 and slowly decaying outer radii, the dimension can approach n−1.

Significance. If correct, Theorem 1.1 would be a clean, exact Hausdorff dimension result for a natural class of annulus limsup sets, with an interesting dimension drop to n−1. The method of deriving a shifted mass transference principle from Cassel's scaling lemma and the Wang–Wu rectangle principle is potentially reusable. No parameters are fitted, and the main results are obtained from external mass transference principles rather than by circular reasoning. However, as detailed below, the proofs of Theorem 1.2 and of a boundary case of Theorem 1.3 contain load-bearing gaps, so the contribution cannot be accepted in its present form.

major comments (3)

[§2, Proof of Theorem 1.2] The displayed containment in the lower-bound proof is false as stated. For n=2, ρ=2, r=ψ(q)/q=1 and p/q=0, the claimed inner ball is B∞((3/(2√2),3/(2√2)), 1/√2). The point x=(1/(2√2),1/(2√2)) lies in that ball, since its max-norm distance from the centre is 1/√2, but ‖x‖∞=1/(2√2)<1 and ‖x‖2=1/2<1, so x lies in the inner ρ-ball as well as the outer max ball. In fact, points of the claimed ball can have max-norm distance up to (5/(2√2))r from p/q, exceeding r, so the ball is not contained in the outer ball either. Since this inclusion is the only step linking the shifted approximation set to the annulus set W_n(ψ,‖·‖,‖·‖_ρ), the lower-bound half of Theorem 1.2 is not proved as written.
[§4, lower-bound proof of Theorem 1.3] The proof assumes the existence of an integer ℓ∈{1,...,n−1} satisfying τψℓ > (1−∑_{i=ℓ+1}^n τψ_i)/ℓ in the case where some τψ_i<1/n. This is not guaranteed. For n=2, τψ=(0.6,0.4), we have ∑τψ_i=1 and τψ_2=0.4<1/2, but the only candidate ℓ=1 gives 0.6 > 1−0.4 =0.6, which is false. More generally, if τψ_1=...=τψ_{n−1}=a>τψ_n and (n−1)a+τψ_n=1, no ℓ≤n−1 satisfies the strict inequality. These parameters satisfy the hypotheses of Theorem 1.3, so the construction of b_i, and hence the entire lower-bound argument for this case, fails. The theorem may still be true, but a different argument is needed for these boundary cases.
[§1.1, Theorem 1.2] The theorem is stated for 'any ρ-norm' without excluding ρ=∞, and the notation in §1.1 explicitly allows ρ∈R+∪{∞}. For ρ=∞, B∞(p/q,r)=Bρ(p/q,r), so the set W_n(ψ,‖·‖,‖·‖_ρ) is empty, whereas the claimed dimension (n+1)/(1+τψ) is positive. The proof only treats 0<ρ<∞. The statement should either exclude ρ=∞ or state the degenerate result separately.

minor comments (3)

[Page 1] The word 'chronogloical' should be 'chronological'.
[§2, Theorem 2.1] Condition (1) is written as a limit superior with 'ψ(q)≠0' under the limit; since ψ is positive, this qualifier is redundant and should be removed.
[§4, upper-bound proof] The covering argument uses the phrase 'standard geometric argument' for the number of balls needed to cover a shifted rectangle; spelling out the one-line estimate would improve readability and make the exponent calculation easier to verify.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the dimension theorems are obtained from external mass-transference principles plus a shifted-scaling lemma; the only self-citation is an auxiliary lemma and is not load-bearing in a circular sense.

full rationale

The central claims (Theorems 1.1–1.3) are not obtained by fitting a parameter to the target set or by defining the target set in terms of the conclusion. Theorem 1.1 is deduced from Theorem 1.3, and Theorem 1.3 is proved by applying the external Wang–Wu mass transference principle from rectangles to rectangles, with Minkowski's theorem supplying the full-measure divergence hypothesis. Theorem 1.2 is reduced to a perturbed approximation statement (Theorem 2.1), whose proof uses Khintchine's theorem and the classical Beresnevich–Velani mass transference principle; the perturbed centres are explicit functions of q, not fitted to the annulus set. The only self-citation with a proof role is Lemma 3.1, taken from the authors' preprint [2] ('Bad is null'); it is a general measure-theoretic scaling lemma used inside the shifted-mass-transference machinery. It is not the target dimension result, and its content is not equivalent to the paper's conclusions. There is therefore no exhibited reduction of a prediction to its own inputs. Separately, the geometric containment claimed in the proof of Theorem 1.2 appears to be false (for example n = 2, rho = 2, r = 1), but a false containment is a correctness defect, not a circularity, so it does not raise the circularity score under the stated rules.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new parameters fitted to data and no invented entities. It relies on established external theorems and on one auxiliary lemma from a preprint coauthored by Ward. The main unproved background assumptions are the mass transference principles and the existence of the integer ell in the rectangular lower-bound proof.

assumptions (6)

standard math Mass Transference Principle of Beresnevich and Velani (Theorem 3.1)
External black box cited as [4]; used to pass from full measure of enlarged balls to Hausdorff measure statements.
standard math Wang-Wu Mass Transference Principle from rectangles to rectangles (Theorem 3.3)
External black box cited as [20]; central tool for the lower bound in Theorem 1.3.
domain assumption Generalised Cassel scaling lemma (Lemma 3.1 from [2])
Auxiliary lemma from a preprint coauthored by Ward; not machine-checked but proof sketched in [2]. Used to equate measures of limsup sets after shifting balls and rectangles.
standard math Minkowski's theorem for systems of linear forms
Used in the lower bound of Theorem 1.3 to guarantee full Lebesgue measure of the unshifted rectangle limsup when the exponents sum to 1.
standard math Khintchine's theorem
Used in the proof of Theorem 2.1 to establish the full-measure condition required by the shifted mass transference principle in the divergence case.
ad hoc to paper Existence of the integer ell in the second case of the Theorem 1.3 lower-bound proof
The proof assumes a largest ell with tau_psi_ell > (1 - sum_{i>ell} tau_psi_i)/ell exists. This fails for valid parameters such as n=3, tau_psi=(0.4,0.4,0.2), sum tau_psi=1, so the case split is incomplete.

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Pith. "Pith review of A note on limsup sets of annuli." pith.science (2026). https://pith.science/paper/Z2UHYUMI

@misc{pith2026250209807,
  author       = {Pith},
  title        = {Pith review of: A note on limsup sets of annuli},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z2UHYUMI}},
  note         = {Machine review of arXiv:2502.09807}
}

abstract

We consider the set of points in infinitely many max-norm annuli centred at rational points in $\mathbb R^{n}$. We give Jarn\'ik-Besicovitch type theorems for this set in terms of Hausdorff dimension. Interestingly, we find that if the outer radii are decreasing sufficiently slowly, dependent only on the dimension $n$, and the thickness of the annuli is decreasing rapidly then the dimension of the set tends towards $n-1$. We also consider various other forms of annuli including rectangular annuli and quasi-annuli described by the difference between balls of two different norms. Our results are deduced through a novel combination of a version of Cassel's Scaling Lemma and a generalisation of the Mass Transference Principle, namely the Mass transference principle from rectangles to rectangles due to Wang and Wu (Math. Ann. 2021).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Hausdorff dimension and exact approximation order in $\mathbb{R}^n$

P . Bandi and N. de Saxcé. Hausdorff dimension and exact ap proximation order in Rn. Preprint arXiv:2312.10255, 2023. 1, 2, 3

work page Pith review arXiv 2023

[2] [2]

Bad is null

V . Beresnevich, S. Datta, A. Ghosh, and B. Ward. Bad is nul l. Preprint arXiv:2307.10109, 2023. 6, 7

work page Pith review arXiv 2023

[3] [3]

Beresnevich, D

V . Beresnevich, D. Dickinson, and S. Velani. Sets of exac t ‘logarithmic’ order in the theory of Diophantine approxim ation. Math. Ann., 321(2):253–273, 2001. 1

work page 2001

[4] [4]

Beresnevich and S

V . Beresnevich and S. Velani. A mass transference princi ple and the Dufﬁn-Schaeffer conjecture for Hausdorff measu res. Ann. of Math. (2), 164(3):971–992, 2006. 2, 6

work page 2006

[5] [5]

Besicovitch

A. Besicovitch. Sets of Fractional Dimensions (IV): On R ational Approximation to Real Numbers. J. London Math. Soc., 9(2):126–131,

work page

[6] [6]

Y . Bugeaud. Sets of exact approximation order by rationa l numbers. Math. Ann., 327(1):171–190, 2003. 1

work page 2003

[7] [7]

Y . Bugeaud. Sets of exact approximation order by rationa l numbers. II. Unif. Distrib. Theory, 3(2):9–20, 2008. 1

work page 2008

[8] [8]

Bugeaud and C

Y . Bugeaud and C. Moreira. Sets of exact approximation or der by rational numbers III. Acta Arith., 146(2):177–193, 2011. 1, 2

work page 2011