{"id":"0bee6a3b-152e-43fa-94ac-85fdc9f7f569","arxiv_id":"2502.09807","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves Jarnik-Besicovitch-type Hausdorff dimension formulas for limsup sets of annuli around rational points, including rectangular and norm-difference annuli.","lead":"This paper computes the Hausdorff dimension of points that lie in infinitely many thin shells around rational points in n-dimensional space. The result extends a classical theorem of Jarnik and Besicovitch and shows that very thin shells can drop the dimension to n minus 1.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's lower-bound proof uses a false geometric containment: the shifted ball is neither contained in the outer max-norm ball nor outside the inner ρ-norm ball; as written, Theorem 1.2 is unproved.","rationale":"The reader's weakest assumption identifies exactly the same issue: the geometric inclusion used in the proof of Theorem 1.2 is false as written. My independent check with n=2, ρ=2, r=1 confirms that a point in the claimed shifted ball is neither in the outer max-norm ball nor outside the Euclidean ball, so the lower-bound embedding into the annulus set collapses. This is the most load-bearing concern because Theorem 1.2 is one of the paper's advertised results, and the flawed containment is the essential bridge from the perturbed approximation machinery to the norm-difference annulus. The theorem may still be true with a different embedding, e.g. placing a small ball near a corner of the max-norm ball, but the submitted proof does not establish it. I also note the reader's second concern about the existence of the integer ℓ in the lower-bound proof of Theorem 1.3; it is real but more localized, occurring when ∑τψi=1 and some exponent is below 1/n, and it appears repairable by allowing equality in the defining inequality. The central equal-exponent Theorem 1.1 is not directly threatened by either flaw, since its lower bound is handled by the first case of Theorem 1.3 and by the separability of the annular geometry. Nonetheless, the paper as a whole claims Theorems 1.2 and 1.3, and those proofs are not valid as written. Therefore I do not change the reader's REJECT verdict: the advertised package is not supported, even though the likely fix is an improved embedding rather than a fundamentally different argument.","tokens_in":14623,"tokens_out":10894,"duration_ms":108353,"concrete_test":"Fix n=2, ρ=2, r=1 and test the displayed containment numerically. For x=(1/(2√2), 1/(2√2)), compute ||x||∞ and ||x||2; both are less than 1, while the distance from x to the claimed shifted center c=(3/(2√2), 3/(2√2)) equals the claimed radius 1/√2. This settles that the containment is false. Then test the natural repair: center a small ball at (1-ε)r in each coordinate with ε=(1-n^{-1/ρ})/4 and radius εr; verify it lies inside B∞ and outside Bρ. If the repair works, Theorem 1.2 is likely recoverable, but the submitted proof requires a corrected construction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single load-bearing flaw is in the proof of Theorem 1.2. The proof asserts the containment B∞(p/q,r) \\ Bρ(p/q,r) ⊇ B∞(p/q + (±r(n^{1/ρ}+n^{-1/ρ})/2, ...), r(n^{1/ρ}-n^{-1/ρ})). This is false. For n=2, ρ=2, r=1, the claimed shifted ball has center c=(3/(2√2), 3/(2√2)) and radius 1/√2. The point x=(1/(2√2), 1/(2√2)) lies in this shifted ball, since its distance from c is exactly 1/√2, but ||x||∞=1/(2√2)<1 and ||x||2=1/2<1, so x is not in B∞(0,1) \\ B2(0,1). The error is not a minor constant slip: the shifted ball's points can reach max-norm distance (3/2)n^{1/ρ}-(1/2)n^{-1/ρ}>1 from p/q, so it is not contained in the outer ball, and moving inward from the shifted center penetrates the inner ρ-ball. Since this containment is the only link between the perturbed approximation theorem and the annulus set, the lower-bound half of Theorem 1.2 is not proved as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14814,"tokens_out":23611,"duration_ms":200175,"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":[{"comment":"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.","section":"§2, Proof of Theorem 1.2"},{"comment":"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.","section":"§4, lower-bound proof of Theorem 1.3"},{"comment":"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.","section":"§1.1, Theorem 1.2"}],"minor_comments":[{"comment":"The word 'chronogloical' should be 'chronological'.","section":"Page 1"},{"comment":"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.","section":"§2, Theorem 2.1"},{"comment":"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.","section":"§4, upper-bound proof"}],"recommendation":"major_revision","confidential_remarks":"The two technical gaps are serious but localized; I do not see a fundamental flaw in the overall strategy. I would encourage the authors to fix the geometric containment in Theorem 1.2 and to add a separate treatment of the boundary case in Theorem 1.3. With those repairs, the paper could be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nBut 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.\n\nTheorem 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.\n\nWhat 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.\n\nWho 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.","headline":"Genuine new result in Theorem 1.1, but two advertised theorems have load-bearing proof gaps; repairable, but not acceptable as is.","tokens_in":15418,"tokens_out":10421,"would_cite":false,"duration_ms":90704,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J83","28A80","11K60"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Hausdorff dimension","Diophantine approximation","limsup sets","annuli","mass transference principle","Jarník–Besicovitch theorem","exact approximation order"],"falsifier":"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.","tokens_in":14310,"feed_emoji":"📐","tokens_out":10549,"duration_ms":95794,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hausdorff dimension of rational-centered annuli pinned down","feed_subtitle":"For power-law shrinking shells, the dimension is a simple minimum; slow outer decay plus fast thinning drives it to n-1.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original mass transference principle used to turn full-measure statements on blown-up balls into Hausdorff-measure statements on the original balls.","marker":"[4]"},{"why":"Provides the mass transference principle from rectangles to rectangles that gives the lower-bound dimension formula for shifted hyperrectangles in Theorem 3.4.","marker":"[20]"},{"why":"Contains the version of Cassel's scaling lemma (Lemma 5.3) and its rectangle analogue (Lemma 5.7) used to absorb centre perturbations in the shifted mass transference principles.","marker":"[2]"},{"why":"Cassel's original scaling lemma, the foundation that Lemma 3.1 generalizes.","marker":"[9]"},{"why":"Khintchine's theorem is used in the proof of Theorem 2.1 to verify the divergence condition that activates the mass transference argument.","marker":"[14]"}],"fun_headline_variants":["Exact dimension for rational-centered annuli","Annuli dimension: min of two rates, thinning wins","Slow outer radius decay, fast thinning: dimension n-1","Power-law annuli: Hausdorff dimension formula","Rational annuli: dimension collapses to n-1 with thin shells"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exact dimension for rational-centered annuli","Annuli dimension: min of two rates, thinning wins","Slow outer radius decay, fast thinning: dimension n-1","Power-law annuli: Hausdorff dimension formula","Rational annuli: dimension collapses to n-1 with thin shells"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000253,"raw_usage":{"total_tokens":1579,"prompt_tokens":971,"completion_tokens":608,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":528}},"tokens_in":587,"tokens_out":608,"duration_ms":6774,"temperature":1.0,"reasoning_tokens":528,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T20:29:16.060157+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Beresnevich and S","cited_arxiv_id":null,"evidence_quote":"Supplies the original mass transference principle used to turn full-measure statements on blown-up balls into Hausdorff-measure statements on the original balls."},{"cited_title":"Wang and J","cited_arxiv_id":null,"evidence_quote":"Provides the mass transference principle from rectangles to rectangles that gives the lower-bound dimension formula for shifted hyperrectangles in Theorem 3.4."},{"cited_title":"Bad is null","cited_arxiv_id":"2307.10109","evidence_quote":"Contains the version of Cassel's scaling lemma (Lemma 5.3) and its rectangle analogue (Lemma 5.7) used to absorb centre perturbations in the shifted mass transference principles."},{"cited_title":"Some metrical theorems in Diophantine appr oximation","cited_arxiv_id":null,"evidence_quote":"Cassel's original scaling lemma, the foundation that Lemma 3.1 generalizes."},{"cited_title":"Einige Sätze über Kettenbrüche, mit Anwe ndungen auf die Theorie der Diophantischen Approximatione n","cited_arxiv_id":null,"evidence_quote":"Khintchine's theorem is used in the proof of Theorem 2.1 to verify the divergence condition that activates the mass transference argument."}],"review_version":1}