arXiv:2503.21180v3
Metric theory of inhomogeneous Diophantine approximations with a fixed matrix
math.NT11J2011J8311J13
Abstract
In this paper we develop a metric theory of inhomogeneous Diophantine approximation for the case of a fixed matrix. We use transference principle to connect uniform Diophantine properties of a pair of a matrix and a vector with the asymptotic Diophantine properties of the transposed matrix , and vice versa, the asymptotic Diophantine properties of a pair with asymptotic Diophantine properties of the transposed matrix. In these setups, we prove analogues of classical statements of metrical homogeneous Diophantine approximations and answer some open questions that were raised in recent works.
Dependence graphs
Proof lineage
Metric theory with a fixed matrix
A statement-restricted graph for the transference, second-moment, lacunary escaping-set, and product-dimension inputs used in the paper. Historical attributions, contextual literature, and arguments reproduced in the focal paper or an included source are excluded; published books are terminal.
Open dependence graph →AI-generated audit
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 findingsCorrect
The central fixed-matrix transference, measure, winning, and Khintchine-type results are correct. Four material arguments are not valid or complete as printed: the lacunarity pigeonhole argument, the metric Cassels overlap count, the local independence normalization, and the claim that different subsequences automatically yield different infinite sums. Explicit repairs below verify the stated conclusions.
Theorems 3.1–3.4Correct
Uniform and asymptotic fixed-matrix transference results
Pages 12–15 and 28–36 · Theorems 3.1–3.4 and their general forms · arXiv:2503.21180v3
The transference inequalities correctly convert homogeneous approximants for into restrictions on the scalar products of a shift with the dual vectors. The Borel–Cantelli argument on non-exceptional affine subspaces proves the null conclusions, and the lacunary hyperplane strategy establishes the HAW conclusions. The local independence ratio in Lemma 6.5 must be normalized by the reciprocal ball measure, as the downstream Chung–Erdős computation already requires; the slicing repair below supplies that normalization and verifies the measure argument.
Theorems 3.6 and 3.7Correct
Metric Kurzweil-type null statements on affine slices
Page 16 and pages 34–39 · Theorems 3.6–3.7 and proofs · arXiv:2503.21180v3
The summability hypothesis removes the scalar-product exceptional set, while the transference inequality turns any too-good inhomogeneous approximation into one of those excluded events. The metric Cassels lemma supplies the complementary full-measure limsup statement. After replacing its one-dimensional overlap count by the coordinatewise -dimensional count below, the second-moment estimate is valid and the advertised affine-slice conclusions follow with the stated constants.
Theorem 3.8 and Theorem 9.1Correct
Khintchine-type comparison and uncountability
Page 17 and pages 39–42 · Theorem 3.8, Theorem 9.1, and proof · arXiv:2503.21180v3
Part (a) follows by subtracting consecutive inhomogeneous best approximants. For part (b), Proposition 5.1 supplies a sufficiently sparse subsequence whose denominator sums and error tails are geometrically dominated, giving the factor . To justify uncountability, choose the candidate errors in consecutive pairs with each candidate at most one sixteenth the norm of its predecessor, and choose one member of every pair. The first pair at which two binary choices differ contributes more than the combined later tails, so the sums are distinct. This repairs the printed implication and proves that there are uncountably many shifts satisfying the estimate.
Proposition 5.1Correct
Uniform growth and decay of best approximations
Pages 23–24 · Proposition 5.1 and proof · arXiv:2503.21180v3
Part (a) is the cited standard growth bound. For part (b), pair the small remainder cubes into antipodal pairs. Among , with , one antipodal pair contains at least remainders. Change signs so all of them lie in the same member of the pair, and partition the signed denominator box into half-size boxes. Two denominators then share a box; their nonzero difference has norm at most and error at most . Hence as stated.
02Proofs5 reported findingsContains incorrect or incomplete proofs
Several proofs require material repair. Proposition 5.1 mixes numerator remainders with full lattice vectors; Lemma 7.9 uses a one-dimensional ceiling in an -dimensional count and then inserts extraneous factors; Lemma 6.5 states the wrong normalization without proving the pair estimate; and Theorem 9.1's uncountability inference is false for arbitrary nonzero summands. Each gap has a verified repair that preserves the theorem statements.
Proposition 5.1(b)Incorrect as written · verified repair
The pigeonhole argument uses incompatible spaces and signs
Pages 23–24 · proof of Proposition 5.1(b) · arXiv:2503.21180v3
The remainders are -vectors, but the proof places them in boxes in . It then adds both signs of selected denominators without ensuring that the signed remainders remain in one small cube. Instead, partition the -dimensional remainder shell into cubes and pair antipodal cubes. One pair contains at least remainders; orient each by a sign into the same cube. Two of the corresponding signed denominators occupy the same one of half-size denominator boxes. Their difference is nonzero because distinct best-approximation indices have distinct norms, and it has exactly the bounds stated in the statements audit. The final printed inequality also has an extra factor on its left; deleting it yields the proposition.
Lemma 6.5Incomplete as written · verified repair
The independence ratio is inverted and its derivation is omitted
Page 27 · Lemma 6.5 · arXiv:2503.21180v3
If is the fixed ball, asymptotic independence relative to normalized measure means not . The reciprocal is also the value used later to make the Chung–Erdős lower bound equal to . To prove it, decompose into slabs for the lower-frequency functional, apply the preceding uniform strip estimate to the higher-frequency functional on every complete slab, and bound the two boundary fragments. The condition makes the total boundary contribution negligible uniformly in the lower-frequency slab. This gives the displayed reciprocal limit and completes the omitted step.
Lemma 7.9Incorrect as written · verified repair
The metric Cassels overlap count must be coordinatewise
Pages 36–37 · proof of Lemma 7.9, especially display (7.27) · arXiv:2503.21180v3
The number of -dimensional grid boxes meeting a fixed cube is estimated by a single ceiling, and the next double sum contains extra factors . Coordinatewise counting instead gives, for sufficiently sparse , at most fine cubes meeting one fixed coarse cube. Since the fine cubes are disjoint, summing their overlap volumes over the coarse cubes yields The Chung–Erdős lemma then gives the claimed full-measure limsup.
Theorem 9.1(b)Incomplete as written · verified repair
Different subsequences need not produce different sums
Page 41 · last paragraph of the proof of Theorem 9.1 · arXiv:2503.21180v3
Nonzero summands alone do not make the map from subsequences to infinite sums injective. Thin the liminf-realizing candidate sequence so every error vector has norm at most one sixteenth of its predecessor and all index-gap requirements remain satisfied. Group candidates in pairs and select one candidate from each pair. At the first differing pair, the norm of the difference of the two selected terms is at least fifteen sixteenths of the larger term, whereas the sum of both later tails is less than two fifteenths of it. The resulting sums are therefore distinct, and the binary choices give continuum many valid shifts.
Exceptional-set and complement notationTypo
Two set-theoretic displays have unique corrections
Pages 25 and 13 · exceptional-set definition and discussion after Theorem 3.2 · arXiv:2503.21180v3
The exceptional set defined by infinitely many occurrences requires the limsup , not a single union. Also, when bad approximability forces only the trivial singular shifts, it is the singular set itself that is countable, not its complement; the complement was just proved HAW. The definitions and adjacent conclusions determine both corrections uniquely.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.