arXiv:2603.25988v1
Uniform Diophantine approximation with restrictions via total density of collections of subspaces
math.NT11J1311J2014N20
Abstract
In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of variables. Throughout the years, this argument has been extensively modified and generalized. Most recently, Kleinbock et al. (2025) introduced a general framework of Diophantine systems and showed that a certain topological property called total density implies a far-reaching generalization of Khintchine's result. We describe a way to establish total density for a variety of Diophantine systems, and thus prove that the sets of singular objects are uncountable and dense in a wide range of set-ups in Diophantine approximation. As a special case, we establish such a result for inhomogeneous approximation, proving the existence of uncountably many singular systems of affine forms with a fixed translation part. One can also consider approximation with prime denominators, or more generally, approximation under some strong restrictions on numerators and denominators.
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Proof lineage
Uniform Diophantine approximation with restrictions via total density of collections of subspaces
A statement-restricted graph for the total-density criteria, their uniform-approximation consequences, the convex-hull criterion, and the separately marked optimality branch. Context, historical attributions, alternative proof sources, and arguments reproduced inside the focal paper are excluded.
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Audit summary
Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.
Current report
Detailed mathematical audit
01Statements4 reported findingsContains wrong statements
The total-density conclusions of Theorems 1.5, 1.6, 2.1, and 2.3 are correct after the verified proof repairs recorded below. Their advertised uniform-approximation consequence is false, however, when the exceptional family is allowed to consist of arbitrary proper analytic submanifolds: the invoked KMWW theorem requires closed nowhere dense exceptional sets. Restricting the claim to proper closed analytic submanifolds repairs every affected application.
Formula (1.6), Theorems 1.5–1.6, and Corollaries 1.8–1.9Incorrect as written · verified scope repair
The analytic exceptional sets must be closed
PDF pages 4–8 · formula (1.6), Theorems 1.5–1.6, and Corollaries 1.8–1.9 · arXiv:2603.25988v1
The paper claims that the relevant uniform set remains uncountable and dense after deleting any countable family of proper analytic submanifolds of , while both its displayed Theorem B and the exact KMWW theorem require the forbidden sets to be closed nowhere dense. Enumerate open balls with rational centers and positive rational radii. Each ball is a proper real-analytic submanifold as a proper open subset, and the resulting countable family covers ; the displayed set difference is therefore empty. Inserting ‘closed’ before ‘analytic submanifolds’ is the verified repair. Proper closed analytic submanifolds are nowhere dense and are respected by the affine resonant fibers, exactly as the paper checks immediately before formula (1.6). The total-density assertions themselves are unaffected.
KMWW, Compositio Mathematica version of record ↗Theorems 2.1 and 2.3Correct
The two geometric criteria yield property (TDS)
PDF pages 9–10 and 23–26 · Theorems 2.1 and 2.3 and their proofs · arXiv:2603.25988v1
For Theorem 2.1, the generic-dimension proposition makes the relevant product families dense, and the repaired local-intersection lemma promotes the prescribed closed pieces to a totally dense subcollection. For Theorem 2.3, the partition-sensitive finite convex-hull lemma reduces the direction sets to finitely many pieces, after which the same total-density proposition applies. Replacing the few unprimed direction sets by the primed sets just constructed and correcting the opposite-ball notation makes every hypothesis match. These repairs establish all three alternatives of Theorem 2.1 and the full convex-hull criterion of Theorem 2.3.
Exact arXiv version 1 ↗Theorem 1.6Correct
Logarithmic density and two asymptotic halfspheres give a totally dense subcollection
PDF pages 5–6 and 11–12 · Theorem 1.6 and proof · arXiv:2603.25988v1
Logarithmic density supplies every required numerator-to-denominator ratio along each limiting direction. For radii tending to infinity, extract convergent pairs of admissible halfspheres. If two limiting supporting spheres differ, Theorem 2.1(a) applies. Otherwise select from every finite pair a halfsphere whose support differs from the common limiting sphere; points on its limit are approached by parameter directions off that whole sphere, and symmetrization gives Theorem 2.1(c). This verified compactness repair makes the dichotomy exhaustive and proves property (TDS). With ‘closed’ inserted in the exceptional-set clause, the two inhomogeneous corollaries follow from the stated augmented-matrix substitutions.
Exact arXiv version 1 ↗Theorem 1.5Correct
The two product-restriction collections are totally dense
PDF pages 4–5 and 35 · Theorem 1.5 and proof · arXiv:2603.25988v1
In part (a), set and choose an -dimensional subspace of the real span of ; the rank inequality guarantees its existence. The normalized parameter closure contains for a -subsphere separated from the fixed direction, so Proposition 3.4 and Lemma 4.2 apply in their stated ranges. In part (b), choose the sign of an unbounded coordinate subsequence and separate that direction from both signs of the fixed direction. Bounded Hausdorff distance supplies all numerator ratios, Corollary 3.6 gives density, and Lemma 4.2 gives total density. Thus both total-density assertions are correct after the repairs.
Exact arXiv version 1 ↗02Proofs6 reported findingsContains incorrect or incomplete proofs
The proofs are not complete as printed. The KMWW reduction is invoked outside its closedness and countability hypotheses; the proof of Theorem 1.6 assumes a non-exhaustive limiting-halfsphere dichotomy; Lemma 4.2 omits mixed coordinate cases; and the proof of Theorem 1.5 uses inadmissible dimensions and swapped normalizations. The first issue forces the statement restriction recorded above. The other defects have verified repairs, and the remaining extrema, primes, opposite balls, signs, and ranks have uniquely determined local corrections.
KMWW reduction behind formula (1.6)Incomplete as written · verified repair after restricting the statement
Two source hypotheses are not discharged in the printed reduction
PDF pages 3–8 · Theorem B and the ‘hence’ clauses following property (TDS) · arXiv:2603.25988v1
First, arbitrary analytic exceptional sets need not be closed or nowhere dense, so KMWW cannot imply the printed scope; this is repaired only by inserting ‘closed’ in every affected conclusion. Second, Theorem B requires a countable aligned collection, whereas the focal criteria allow arbitrary subsets and . The latter omission has a verified repair. In the second-countable space , start with a countable subfamily whose union is dense. Recursively, for every selected fiber and basis open set meeting , total density supplies a nonempty open set inside the closure of the eligible fibers meeting ; select countably many eligible fibers with union dense in that open set. The union over countably many stages is a countable totally dense subcollection. Finally, all relevant affine fibers have no isolated points, as checked in the Sources dimension, so the repaired uncountability step applies.
KMWW, Compositio Mathematica version of record ↗Proof of Theorem 1.6Incomplete as written · verified repair
The two printed limiting-halfsphere cases are not exhaustive
PDF page 12 · proof of Theorem 1.6 · arXiv:2603.25988v1
Compactness gives a limiting halfsphere, but the proof neither derives its two declared cases nor justifies replacing that halfsphere by its entire supporting sphere. Choose radii and admissible pairs of halfspheres at those radii, then pass to limits of both supports and orientations. If two limiting supports differ, Theorem 2.1(a) applies. Otherwise all selected supports converge to one -sphere . From each finite pair choose a halfsphere whose support is not and pass to a limiting halfsphere in . Every point of that limit is approached by directions with norm exceeding and lying outside ; symmetrization supplies the opposite halfsphere. Consequently which is precisely Theorem 2.1(c). This proves the missing dichotomy without changing Theorem 1.6.
Exact arXiv version 1 ↗Lemma 4.2Incorrect as written · verified repair
The printed cases omit mixed coordinate separation
PDF pages 21–22 · proof of Lemma 4.2 · arXiv:2603.25988v1
A parameter outside the two product balls need not have its direction coordinate separated from both signs or its numerator coordinate separated from both signs; one sign can be excluded by each coordinate. Fix . For the max product metric used by the printed case split, if the direction is not already separated from , exclusion from the ball about forces ; for any equivalent standard product metric the same argument works after replacing by a fixed smaller multiple. For and , respectively, use and Since and , both give in the max-metric normalization. The angle lemma now applies sign by sign and proves the required local intersection.
Exact arXiv version 1 ↗Proof of Theorem 1.5Incorrect as written · verified repair
The normalized parameters and the sphere dimension are outside the printed ranges
PDF page 35 · proof of Theorem 1.5 · arXiv:2603.25988v1
The proof fixes a pair in the wrong coordinate order, writes the normalization with the two coordinates reversed, and applies Proposition 3.4 with and the full span of , even when or that span has dimension larger than . It also calls an -subsphere, which is the whole and cannot satisfy the displayed positive-separation condition. Fix and In part (a), take , choose of dimension inside , choose a -subsphere away from the fixed direction, and use with a positive scaling integer . The parameter closure then contains times that subsphere and Proposition 3.4 applies. In part (b), take the separation radius below half the distance from the limiting coordinate direction to each of and . These changes verify both branches.
Exact arXiv version 1 ↗Proposition 3.5 and Lemma 3.2Typo
The separating functional needs a supremum and the constructed matrix has rank
PDF pages 18–20 · Lemma 3.2 and converse proof of Proposition 3.5 · arXiv:2603.25988v1
Strict separation gives a functional with uniformly over the closed direction set. Thus the open set in Proposition 3.5 is defined by and the positive constant is ; the two printed infima contradict the inequality used on the same page. In Lemma 3.2, adjoining new image vectors to the fixed vectors makes the displayed augmented matrix have rank , not rank . Both corrections are mechanically fixed by the surrounding displays and alter no conclusion.
Exact arXiv version 1 ↗Proposition 4.3 and proof of Theorem 2.3Typo
Opposite balls and the newly constructed primed direction sets are mistyped
PDF pages 22–26 · Proposition 4.3 and proof of Theorem 2.3 · arXiv:2603.25988v1
Every repeated pair in Proposition 4.3 has second member . In the proof of Theorem 2.3, after the recurrent set and the complementary finite set are defined, the proof must remove and build from , rather than reverting to . With those uniquely determined replacements, the dense collections have exactly the hypotheses established in equations (5.4)–(5.8).
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.
04Sources4 reported findingsContains incorrect or incomplete source use
The exact KMWW theorem, Eggleston’s Carathéodory theorem, and Boyd–Vandenberghe separation result are sufficient for their intended mathematical roles. The focal affine fibers immediately satisfy the repaired no-isolated-points condition needed for KMWW uncountability. The source use is nevertheless incomplete as printed because the paper applies KMWW beyond its closed-exceptional-set and countable-family hypotheses; countability has a verified skeleton repair, while closedness requires narrowing the stated conclusions.
KMWW Theorem 1.5 — uncountability hypothesisApplicable after verified source correction
Every focal affine fiber discharges the repaired no-isolated-points condition
PDF pages 3–8 and 35 · Theorem B and every focal uniform-approximation application · arXiv:2603.25988v1
The exact Compositio theorem says that uncountability follows when each aligned set is either connected or has no isolated points, but the VOR audit gives a connected-singleton counterexample and verifies the corrected sufficient hypothesis ‘has no isolated points’. The focal paper repeats the invalid connectedness alternative in Theorem B. Its actual uses do not enter the bad regime: in an ambient every resonant set is an affine space of dimension . The hypotheses of Theorem 1.5 force , and the range in Theorem 1.6 does the same. Corollary 1.8 applies that theorem to augmented matrices with , while Corollary 1.9 assumes . Hence every aligned fiber in every proof-critical application has positive dimension and no isolated points. This immediate dimension check fully discharges the repaired KMWW condition.
KMWW, Compositio Mathematica version of record ↗KMWW Theorem 1.5 — closedness and countabilityIncorrect or incomplete source use
The focal ‘hence’ clauses exceed two explicit source hypotheses
PDF pages 3–8 · formula (1.6), Theorems 1.5–1.6, and Corollaries 1.8–1.9 · arXiv:2603.25988v1
KMWW requires a countable aligned collection and a countable family of closed forbidden sets respected by that collection. The focal paper allows arbitrary and in several criteria and claims avoidance of arbitrary proper analytic submanifolds. A countable totally dense skeleton can be extracted by the recursive second-countability construction given in the Proofs dimension, so the first mismatch is repaired without changing a conclusion. The second mismatch cannot be suppressed: a countable cover by proper open analytic submanifolds makes the claimed complement empty. Restricting the exceptional family to proper closed analytic submanifolds exactly matches the source and repairs the applications.
KMWW, Compositio Mathematica version of record ↗Eggleston, Chapter 2.2Applicable and sufficient
Carathéodory’s theorem is quoted with the correct dimension bound
PDF page 24 · Theorem C and proof of Theorem 5.1 · arXiv:2603.25988v1
The recalled result is the exact finite-dimensional Carathéodory theorem specialized to the origin: if , then some subset of at most points already has the origin in its convex hull. The focal proof applies it once to obtain a base witness and at most once for each base witness lying in . Their union has cardinality at most and satisfies the stated deletion property. No stronger conclusion is attributed to the book, and the remainder of the partition-sensitive argument is proved internally.
H. G. Eggleston, Convexity, Chapter 2.2 ↗Boyd–Vandenberghe, Section 2.5.1Applicable and sufficient
Strict separation applies to the compact convex hulls used in both branches
PDF pages 19 and 29–30 · Proposition 3.5 and the optimality proof · arXiv:2603.25988v1
The relevant direction set lies on the unit sphere, so its closure is compact and its convex hull is compact. If that convex hull omits the origin, strong separation from the singleton supplies a functional with a uniform strict sign. In Proposition 3.5 this yields and an open matrix set uniformly separated from every associated affine fiber; in the optimality proof it yields the printed positive margin . Thus the cited theorem has exactly the hypotheses and strength required. The infimum in Proposition 3.5 is an internal notation typo, not a deficiency in the source.
Boyd and Vandenberghe, Convex Optimization ↗