Published paper

Singularity, weighted uniform approximation, intersections and rates

Dmitry Kleinbock, Nikolay Moshchevitin, Jacqueline M. Warren, Barak Weiss

MSC 2020

11J1311H0611J5411J82

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Publication metadata

Journal citation: Compositio Mathematica 161(11), 2990–3016 (2025)
Publication date: 2026-01-08
DOI: 10.1112/S0010437X25102777 ↗
arXiv counterpart: arXiv:2409.15607v4
MSC 2020: 11J13, 11H06, 11J54, 11J82
As specified in the publication (MSC 2020): 11J13, 11H06, 11J54, 11J82 ↗Version-of-record HTML metadata (primary 11J13; secondary 11H06, 11J54, 11J82)
Publication license: CC BY 4.0 ↗

Role in dependence graphs

Proof-critical source

Uniform Diophantine approximation with restrictions via total density of collections of subspaces

This paper is included only for the following marked statement:

Theorem 1.5 · Compositio Math. 161 (2025), theorem in §1.3; proof in the axiomatic-theorem sectionConverts a totally dense aligned collection into uncountably many dense uniformly approximable points outside any prescribed countable family of proper analytic submanifolds.

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Version of record · Compositio Mathematica 161(11), 2990–3016 (2025)

Dmitry Kleinbock, Nikolay Moshchevitin, Jacqueline M. Warren, Barak Weiss. Singularity, weighted uniform approximation, intersections and rates. Compositio Mathematica 161(11), 2990–3016 (2025).

Open audited source ↗

Generated August 23, 2026

01Statements5 reported findingsContains wrong statements

The density conclusion in Theorem 1.5(a) and the principal Diophantine applications are correct after the proof repairs recorded below. Theorem 1.5(b), however, is false under its printed abstract hypotheses because a connected member of the aligned family may be a singleton. The forbidden-set scope of Theorem 3.4 is also false without the word ‘closed’. The reversed monotonicity word in Theorems 1.13 and 7.2 is a mechanical typo.

Theorem 1.5(a)Correct

The abstract density conclusion is correct

Journal pages 2994 and 2998–2999 · Theorem 1.5(a) and Steps 1–3 of its proof · version of record

Starting in an arbitrary relatively compact open set, total density selects an aligned resonant set meeting the current neighborhood and escaping the next forbidden closed set. Respect and continuity permit a smaller open set whose closure remains inside the previous one and on which every earlier approximation inequality holds at the next height. The nested compact-closure argument gives a common point, and monotonicity of each approximating function supplies the required inequality for every sufficiently large parameter. All countably many systems and forbidden sets are visited.

Cambridge version of record ↗

Theorem 1.5(b)Incorrect

Connected singleton resonant sets contradict the claimed uncountability

Journal page 2994 · Theorem 1.5(b); journal page 2999 · Step 4 · version of record

Let $Y=\mathbb{N}$ with the discrete metric, let $\mathcal L=\{\{j\}:j\in\mathbb{N}\}$ , and let $\mathcal R=\varnothing$ . For every $k$ , take $X_k=Y$ , $\varphi_k=\operatorname{Id}$ , indices $s\in\mathbb{N}$ , $d_{k,s}(x)=0$ when $x=s$ and $d_{k,s}(x)=1$ otherwise, heights $h_{k,s}=s+1$ , and $f_k(t)=1/2$ . The space is locally compact, all distance functions are continuous, $\mathcal L$ is totally dense relative to the empty collection, respects it, and is aligned with every system. Every $L\in\mathcal L$ is connected. Yet every point of $Y$ is uniform by reusing its zero-distance index, so (1.11) equals the countable set $Y$ , contradicting part (b). Step 4 fails at exactly the assertion that adjoining the countable output set preserves (1.8): when $L=\{y\}$ , no aligned singleton meeting $L$ can escape $\{y\}$ . Requiring every $L$ to have no isolated points repairs the proof and covers all positive-dimensional families used in Sections 3–7, but the printed abstract statement is false.

Cambridge version of record ↗

Theorem 3.4Incorrect as written · verified repair

The forbidden analytic submanifolds must be closed

Journal page 3002 · statement of Theorem 3.4 · version of record

The theorem quantifies over an arbitrary countable collection of proper analytic submanifolds, omitting the closedness hypothesis used by Theorems 1.5 and 3.3. As written, take a countable cover of $M_{m,n}(\mathbb{R})$ by proper open balls: every ball is a proper real-analytic submanifold, their union is the whole space, and the displayed set is empty rather than dense and uncountable. Inserting ‘closed’ before ‘analytic submanifolds’ is the verified repair; it is exactly the hypothesis invoked by the preceding theorem and by the proof's appeal to the same argument.

Cambridge version of record ↗

Theorems 1.7–1.12Correct

The weighted, manifold, algebraic, and sumset conclusions survive the abstract correction

Journal pages 2995–2997 and 3002–3011 · Theorems 1.7–1.12 and Sections 3–7 · version of record

Every aligned set used for these uncountability conclusions has no isolated points: rational matrix fibers have positive dimension when $n>1$ , the manifold slices have dimension at least one, the product-fractal slices are perfect, and the intersections of two hyperplanes in the sumset argument have dimension at least one when $n\geq3$ . Thus the corrected form of Theorem 1.5(b) applies. The weighted and polynomial-system implementation defects recorded in Part 2 have explicit verified repairs that preserve the stated conclusions and every parameter range.

Cambridge version of record ↗

Theorems 1.13 and 7.2Typo

The monotonicity hypothesis should say non-increasing

Journal pages 2997 and 3011 · Theorems 1.13 and 7.2 · version of record

Both statements call $f$ non-decreasing. Replace ‘non-decreasing’ by ‘non-increasing’. The proof explicitly needs $f$ non-increasing in order that the transferred error function $h$ be non-increasing, and the exponent consequence uses decreasing power functions. The intended correction is uniquely fixed by the proof and changes no later argument.

Cambridge version of record ↗

02Proofs5 reported findingsContains incorrect or incomplete proofs

The nested construction proving Theorem 1.5(a), the incidence and analytic-continuation arguments, the sumset construction, and the convex-body transference calculation are correct. Step 4 cannot prove the printed uncountability theorem, Equation (3.4) encodes approximation for the most favorable weight instead of every weight, and the polynomial encoding in Theorem 5.1 omits the constant coefficient from the height. The latter two defects have verified repairs that preserve the applications.

Proof of Theorem 1.5(a)Correct and complete

The nested topological construction is correct and complete

Journal pages 2998–2999 · Steps 1–3 of the proof of Theorem 1.5 · version of record

The closure of each chosen open set is nonempty and lies inside its predecessor, so local compactness supplies a nonempty common intersection. At stage $\ell$ , the construction records aligned indices for the first $\ell$ systems, increases the common height threshold, and uses continuity at the previous resonant set to impose the next approximation bound. If $T_\ell\leq T<T_{\ell+1}$ , the index from stage $\ell$ has height at most $T$ and its distance is below $f_k(T_{\ell+1})\leq f_k(T)$ . The forbidden sets are avoided one by one.

Cambridge version of record ↗

Proof of Theorem 1.5(b)Incorrect as written

Adjoining the alleged countable output need not preserve total density

Journal page 2999 · Step 4 of the proof of Theorem 1.5 · version of record

Step 4 asserts that $\mathcal R\cup\{\{y_j\}:j\in\mathbb N\}$ still satisfies (1.8). Connectedness alone does not imply this: a connected $L$ may be the singleton $\{y_j\}$ , in which case every $L'$ meeting $L$ at that point can likewise be trapped in $\{y_j\}$ . The discrete counterexample in Part 1 realizes this obstruction and disproves the printed conclusion. Repair classification: Verified repair after strengthening the hypothesis to require that every $L$ have no isolated points; then $L$ itself meets the current open set and is not contained in any singleton, so (1.8) persists.

Cambridge version of record ↗

Theorem 3.4, Equation (3.4)Incorrect as written · verified repair

The infimum over weights does not encode simultaneous weighted approximation

Journal page 3002 · Equation (3.4) and proof of Theorem 3.4 · version of record

The printed distance is $d_{k,(p,q)}(A)=\inf_{\omega\in W_k}\lVert Aq-p\rVert_\alpha$ . Hence $d\leq f_k(t)$ controls only the most favorable weight, whereas membership in $\mathrm{UA}_{m,n}(f_k,W_k)$ requires the same $(p,q)$ to satisfy the inequalities for every $\omega\in W_k$ . For example, with one residual coordinate equal to $0.01$ , weights $\alpha=1$ and $\alpha=2$ , and $f_k(t)=0.05$ , the printed infimum is $0.01$ but the second quasi-norm is $0.1>0.05$ . A verified repair is to use $g_k(t)=\min\{f_k(t),1/2\}$ and the distance $\widetilde d_{k,(p,q)}(A)=\min\{1,\sup_{\omega\in W_k}\lVert Aq-p\rVert_\alpha\}$ , retaining the printed supremum height. Indeed, if $a_{k,i}=\sup_{\omega\in W_k}\alpha_i$ , then on the region $|r_i|<1$ for every residual coordinate one has $\widetilde d(r)=\max_i|r_i|^{1/a_{k,i}}$ , while $\widetilde d(r)=1$ once some $|r_i|\geq1$ . Thus (3.3) makes $\widetilde d$ finite and continuous, with zero set $Aq=p$ ; its height is finite by the lower bound on the $\beta_j$ . The inequalities $\widetilde d\leq g_k(t)<1$ and $h\leq t$ give every weighted inequality required in $\mathrm{UA}_{m,n}(f_k,W_k)$ . Applying Theorem 1.5 to these repaired systems therefore produces a dense uncountable subset of the printed target, preserving both conclusions of Theorem 3.4.

Cambridge version of record ↗

Proof of Theorem 5.1Incorrect as written · verified repair

The standard linear-form height omits the polynomial's constant coefficient

Journal pages 3006–3007 · definition preceding Theorem 5.1 and its proof · version of record

The paper defines $H(P)$ as the maximum of all coefficients of $P$ , but maps $x$ to the nonconstant monomials and then uses the standard linear-form system, whose height is only the norm of the nonconstant coefficient vector $q$ . Its assertion of equivalence therefore omits the constant coefficient $p$ : a relation with $\lVert q\rVert\leq t$ may have $|p|>t$ , so the associated polynomial need not satisfy $H(P)\leq t$ . Replace the standard height by $h_{p,q}=\max\{|p|,\lVert q\rVert\}$ while keeping the same affine-hyperplane distance. The zero sets and alignment used by Theorem 1.5 are unchanged, and for $P(x)=q\cdot\varphi_k(x)-p$ with $q\neq0$ the repaired bound $h_{p,q}\leq t$ is precisely $H(P)\leq t$ . Theorem 1.5 applied with these repaired heights therefore supplies a dense uncountable subset of the printed target, which is sufficient for Theorems 5.1, 5.2, and the simultaneous $k$ -singularity consequence.

Cambridge version of record ↗

Sections 4, 6, and 7Correct and complete after the stated monotonicity repair

The manifold, sumset, and transference arguments are otherwise complete

Journal pages 3003–3011 · Sections 4, 6, and 7 · version of record

Local graph coordinates produce aligned rational slices of positive dimension, analytic continuation establishes the respect property, and the translated pair-of-hyperplanes family is totally dense relative to affine hyperplanes when $n\geq3$ . In Lemma 7.1, the covolume identity, section-volume estimates, and Minkowski bounds yield a nonzero dual integer point with the required constants. After replacing ‘non-decreasing’ by ‘non-increasing’, the scale $\tau(t)$ increases to infinity, the transferred function $h$ is non-increasing, and Theorem 4.2 gives the claimed column-vector rate.