arXiv:2609.36411v1
Twisted Diophantine Approximation I: Asymptotic Theory
math.NT11J8311J20
Abstract
We establish an exact zero-one law for twisted Diophantine approximation with an arbitrary fixed real matrix and a positive non-increasing approximation function satisfying dyadic regularity. The criterion is the convergence or divergence of a dyadic series whose summands are Khintchine-Groshev block volumes divided by homogeneous lattice-point counts. An equivalent formulation is using all successive minima of the associated diagonal lattice trajectory, capturing homogeneous clustering in every direction. We also obtain an almost-sure zero-infinity law for inhomogeneous -Lagrange constants, separate Hausdorff-measure criteria, and dimension results. Within this regularity class, our theorem recovers the Kurzweil and Fuchs-Kim criteria. At the critical exponent, this series determines whether the set of badly approximable shifts has zero or full Lebesgue measure.
Dependence graphs
Proof lineage
Twisted Diophantine Approximation I: Asymptotic Theory
Full statement-restricted proof-dependence graph for arXiv:2609.36411v1. The fixed-matrix zero-one theorem, successive-minima formulation, Lagrange and bad-target classifications, Hausdorff consequences, scalar comparison, and the full quasi-independence chain were checked. The proof-critical external branches close at Henk's lattice estimates and their terminal geometry-of-numbers background, Dani's published correspondence theorems, the general Beresnevich-Velani mass transference theorem, the exact Fuchs-Kim and Beresnevich-Haynes-Velani scalar estimates, and three terminal books for Mahler compactness, generalized Borel-Cantelli, and Hausdorff-Cantelli. Historical, motivational, comparison-only, and results fully reproved in the focal manuscript are excluded.
Open dependence graph →Proof lineage
Twisted Diophantine Approximation II: Uniform Theory
A statement-restricted recursive graph for the proof-critical inputs behind the pressure formula, the no-dimension-drop theorem, the Hausdorff-measure zero-full law, the sharp escape-of-mass construction, and the transference applications. Contextual comparisons, historical attributions, and results merely recovered by the focal paper are excluded. Books are terminal sources.
Open dependence graph →AI-generated audit
Audit summary
Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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
01Statements5 reported findingsNo statement issues found
The exact zero-one law, its successive-minima form, the zero-infinity and critical bad-target classifications, the nonsingular consequence, the Hausdorff-measure and power-law dimension results, and the scalar equivalence with the Fuchs-Kim criterion are correct under the printed dyadic-regularity hypotheses.
Theorem 1.1 and Corollary 1.4No issue found
The fixed-matrix zero-one criterion and its successive-minima form are correct
PDF pages 4-5 and 14-28 · Theorem 1.1, Corollary 1.4, Lemmas 5.1-5.3, Proposition 6.2, and Proposition 7.3 · arXiv:2609.36411v1
For every fixed real matrix and every non-increasing dyadically regular approximation function, the proof identifies the summand with the measure of a disjoint Vitali packet up to constants. Convergence follows from a direct cover. In the divergent case the mixed-scale estimate, fixed-sublattice summation, and incidence bound give quasi-independence on average; the generalized Borel-Cantelli inequality gives positive measure and the independently proved fixed-matrix zero-one law upgrades it to full measure. Rescaling the mixed box to a cube gives the stated product of truncated successive minima with the correct parameters and .
Exact manuscript, arXiv:2609.36411v1 ↗Corollary 2.1 and Proposition 2.2No issue found
The Lagrange zero-infinity law and critical classification follow with the stated quantifiers
PDF pages 9-10 · Corollary 2.1 and Proposition 2.2 · arXiv:2609.36411v1
Lemma 5.3 makes the series criterion invariant under every fixed positive rescaling of . Countable intersections of the divergent full-measure sets force the normalized liminf to be zero, while the complements of all convergent rescalings force it to be infinite. At , the flow normalization gives and , so the series is comparable term by term to and the null-or-conull conclusions for all three bad-target sets follow.
Exact manuscript, arXiv:2609.36411v1 ↗Corollary 2.4No issue found
Nonsingularity forces divergence of the critical series
PDF page 10 · Corollary 2.4 · arXiv:2609.36411v1
Dani's correspondence identifies nonsingularity with failure of forward-orbit divergence. Hence the orbit returns to a compact set at arbitrarily large times. Mahler compactness gives a uniform positive lower bound for its first minimum on those returns, and bounded distortion over the fixed sampling interval transfers that bound to infinitely many sampled times. Each corresponding is bounded below by a fixed positive constant, so the critical series diverges.
Dani, exact 1985 version of record ↗Corollaries 3.1 and 3.2No issue found
The Hausdorff criteria and power-law dimension formula are correct
PDF pages 11-12 · Corollaries 3.1 and 3.2 · arXiv:2609.36411v1
The convergence implication uses the same Vitali cover as Theorem 1.1 with the -cost of each selected ball. For divergence, applying Theorem 1.1 to makes the enlarged-ball limsup conull, and the general mass transference principle on the flat torus yields infinite -dimensional Hausdorff measure for the original limsup. Substitution of then gives the lower bound at and the elementary convergent cover gives every upper bound , proving under the displayed critical-series hypothesis.
Beresnevich-Velani mass transference source ↗Proposition 4.1No issue found
The new dyadic series is equivalent to the scalar Fuchs-Kim series
PDF pages 12-14 · Proposition 4.1 · arXiv:2609.36411v1
For , best-approximation separation gives the required upper bound on the homogeneous cluster count and hence a lower bound for . In the reverse direction, the explicit multiples of counted in Beresnevich-Haynes-Velani equation (6.9) give the required lower cluster count. Fuchs-Kim Lemma 2.1 supplies the summability of from convergence of the block-minimum series, and dyadic regularity sums the remaining terms geometrically. Both implications of the displayed convergence equivalence follow.
Fuchs-Kim, exact arXiv:1501.04714v1 ↗02Proofs4 reported findingsNo proof issues found
The central proofs are correct and complete. The lattice-counting normalization, fixed-matrix zero-one decomposition, Vitali cardinality estimates, mixed-scale packing, exterior-product formula, fixed-sublattice geometric summation, bounded-incidence argument, and all application deductions close with their printed hypotheses. Every proof-critical cited input checked below is used in its valid range.
Sections 5 and 7No issue found in the reviewed proof
The lattice counts and Vitali mass estimates are correct and complete
PDF pages 14-21 · Lemmas 5.1-5.3 and 7.1-7.2 and the convergence proof · arXiv:2609.36411v1
Henk's sublattice comparison gives fixed-dilation control, and his successive-minima upper bound combines with the manuscript's independent-vector lower bound to give the two-sided cube count. The diagonal rescaling is algebraically exact. In Lemma 7.1, the lower Vitali count follows by charging orbit points to selected triple balls, while the upper count uses disjoint translated lattice boxes inside one ambient box; both comparisons preserve uniform constants. These estimates give the Lebesgue and Hausdorff convergence arguments and the correct one-scale masses for divergence.
Henk, Successive Minima and Lattice Points, arXiv:math/0204158v1 ↗Section 6No issue found in the reviewed proof
The fixed-matrix zero-one proof handles every escaping-coordinate pattern
PDF pages 16-18 · Lemma 6.1 and Proposition 6.2 · arXiv:2609.36411v1
Every infinite sequence of distinct integer solutions has a subsequence whose coordinates are either fixed or escape with fixed signs, yielding a countable decomposition. For a component pattern, all centers lie in a coset of the closed subgroup generated by the escaping columns. A proper coset is null; if the subgroup is dense, decrementing one escaping coordinate gives one-sided invariance under each generator. Translation invariance turns those inclusions into invariance modulo null sets, and the Fourier argument forces the component event to be null or conull. The countable union therefore has measure zero or one.
Exact manuscript, arXiv:2609.36411v1 ↗Sections 8 and 9No issue found in the reviewed proof
The quasi-independence-on-average argument closes at every rank
PDF pages 22-28 · Lemmas 8.1-8.2, 9.1-9.3, and proof of Proposition 7.3 · arXiv:2609.36411v1
The local packing lemma bounds later selected balls meeting a fixed earlier ball by a ratio of mixed and one-scale lattice counts. Lemma 9.1 correctly converts each mixed mass to the largest exterior-coordinate term of its active primitive sublattice. For a fixed proper sublattice, the two ordinary geometric arms and the two residual endpoint arms partition all pairs and charge only endpoint masses. Nested mixed boxes form two sublattice chains at each endpoint; equal ranks force equal spans and hence equal primitive sublattices, so at most proper sublattices charge one index. Full-rank pairs contribute quadratically and proper-rank pairs linearly, giving Proposition 7.3 and completing divergence.
Exact manuscript, arXiv:2609.36411v1 ↗Parts I and II source applicationsNo issue found in the reviewed proof
The Dani, mass-transference, scalar-counting, and Borel-Cantelli inputs are applied correctly
PDF pages 5, 10-14, and 20-21 · Remark 1.5, Corollaries 2.4 and 3.1, Proposition 4.1, Lemma 7.2, and the divergence proof · arXiv:2609.36411v1
Dani Theorems 2.14 and 2.20 have the required divergence and bounded-orbit correspondences. The general mass transference theorem applies to the torus with and . Fuchs-Kim's convergence estimate and Beresnevich-Haynes-Velani's elementary count have exactly the scalar hypotheses used. The generalized Borel-Cantelli lower bound and Hausdorff-Cantelli covering lemma are invoked in their standard probability and metric-measure ranges. No proof depends on a stronger conclusion than its source provides.
Exact source bundle for the focal proof chain ↗03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.
04Sources5 reported findingsNo source-use issues found
Every proof-critical external input was located in an exact public version or published source and is sufficient for the use made of it. Henk supplies both lattice-count estimates and a proof of Minkowski's second theorem; Dani supplies the two dynamical correspondences; Beresnevich-Velani supplies the general mass transference theorem; Fuchs-Kim and Beresnevich-Haynes-Velani supply the scalar estimates; and the two cited books supply the probability and Hausdorff covering lemmas. Mahler compactness is represented by a standard terminal book source in the dependency graph.
Henk Lemma 2.1, Theorems 1.3 and 1.5Applicable and sufficient
The lattice-counting source gives exactly the required comparisons
Focal PDF pages 14-15 and 24-26 · Lemmas 5.1 and 9.1 · source arXiv:math/0204158v1 · audited focal arXiv:2609.36411v1
Henk Lemma 2.1 compares the lattice-point count of a convex body with the count in a fixed-index sublattice and yields the focal fixed-dilation bound when the sublattice is . Henk Theorem 1.5 gives the product upper bound in terms of successive minima, while the focal paper supplies its own lower bound. Henk Theorem 1.3 states Minkowski's second theorem, and Section 3 gives the proof used for the covolume-product comparison in Lemma 9.1. The exact source archive and PDF were checked.
Henk, arXiv:math/0204158v1 ↗Dani Theorems 2.14 and 2.20Applicable and sufficient
The exact published proofs support both orbit correspondences
Focal PDF pages 5 and 10 · Remark 1.5 and Corollary 2.4 · Dani printed pages 63-65 · audited focal arXiv:2609.36411v1
The exact 1985 version-of-record scan was inspected. Theorem 2.14 proves that singular systems are precisely those whose associated forward diagonal trajectories diverge, and Theorem 2.20 proves the bounded-trajectory characterization of badly approximable systems. The focal block convention differs only by the standard block permutation and transpose convention. The journal's correction notice does not amend either theorem.
Dani, version of record at EuDML ↗Beresnevich-Velani Section 6.1, Theorem 3Applicable and sufficient
The general mass transference theorem applies on the flat torus
Focal PDF page 11 · proof of Corollary 3.1 · source Section 6.1, Theorem 3 · audited focal arXiv:2609.36411v1
The source theorem is stated for a locally compact metric space with a doubling dimension function and comparable ball measure. The torus with the supremum metric has these properties for . With , the required quotient is monotonic and the transformed radius is , exactly the focal enlarged target. The radii tend to zero and Theorem 1.1 supplies conullity of the enlarged limsup, so every source hypothesis is met.
Beresnevich and Velani, Annals of Mathematics 164 (2006) ↗Fuchs-Kim Lemma 2.1 and Beresnevich-Haynes-Velani equation (6.9)Applicable and sufficient
The scalar convergence and homogeneous-counting inputs are exact
Focal PDF pages 12-14 · Proposition 4.1 · source arXiv:1501.04714v1, Lemma 2.1 proof, and arXiv:1511.06862v2, Section 6.1 equation (6.9) · audited focal arXiv:2609.36411v1
In the Fuchs-Kim proof, convergence of the block-minimum series rules out the large-endpoint case and gives bounded by two adjacent block sums, hence summability over . Beresnevich-Haynes-Velani equation (6.9) counts the positive multiples satisfying both and ; adjoining zero and negative multiples gives precisely the focal lower bound. The continued-fraction inequality then produces the printed comparison.
Beresnevich-Haynes-Velani, exact arXiv:1511.06862v2 ↗Sprindzuk Lemma 5 and Bernik-Dodson Lemma 3.10Applicable and sufficient
The terminal book lemmas have exactly the strengths used
Focal PDF pages 20-21 · convergence proof, Lemma 7.2, and divergent-case proof · arXiv:2609.36411v1
Sprindzuk's second-moment form of the divergent Borel-Cantelli lemma gives the displayed lower bound for the limsup measure from the finite intersection sums. Bernik-Dodson's Hausdorff-Cantelli lemma gives zero Hausdorff measure when the sum of the covering diameters to the power converges. The focal probability space is normalized Lebesgue measure on the torus, and the selected covering diameters tend to zero, so both applications meet their stated hypotheses. These published books are terminal under the dependency-graph stopping rule.