Proof-critical dependence graph

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.

Graph scope12 nodes11 proof-critical linksChecked October 23, 2026

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Proof-critical links

Geometry of Numbers→Successive Minima and Lattice PointsHeadline lineageSource-chain endpoint

Henk imports the elementary lattice normalization, basis-extension, finite-index, and successive-minima span facts before the substantive internal counting argument.

Citation location: Henk Section 2 opening paragraph and equations (2.1)–(2.2), citation [GL87].

Lemma 2.1 and Theorem 1.5Supply the fixed-dilation comparison and the successive-minima upper lattice-count product used in companion-paper Lemma 5.1.

Twisted Diophantine Approximation I: Asymptotic Theory→Twisted Diophantine Approximation II: Uniform TheoryHeadline lineageSource link checked

The focal paper cites companion Lemma 5.1 for both fixed-dilation stability of the lattice count and its comparison with the product of active successive minima. These are exactly the inputs used to prove the profile count and every subsequent transition estimate.

Citation location: Focal Lemmas 4.1–4.2, PDF pp. 22–23; companion Lemma 5.1, PDF pp. 14–15

Lemmas 4.1–4.2The translated-box and successive-minima lattice counts that turn the homogeneous profile into the branching estimates used throughout the covering constructions.

Successive Minima and Lattice Points→Twisted Diophantine Approximation I: Asymptotic TheoryHeadline lineageSource link checked

Henk Lemma 2.1 gives the coset-count comparison used for a fixed dilation, while Theorem 1.5 gives the upper product bound in terms of successive minima. The companion paper supplies the matching lower bound internally by integer combinations of minimizing vectors.

Citation location: Companion Lemma 5.1; Henk Lemma 2.1 and Theorem 1.5

Lemma 5.1Compares a lattice count in a centrally symmetric box with the product of the active successive-minima factors, uniformly under a fixed dilation.

A Mass Transference Principle and the Duffin--Schaeffer conjecture for Hausdorff measures→Twisted Diophantine Approximation II: Uniform TheoryHeadline lineageSelf-contained endpoint

The focal lower construction assigns equal mass down its nested tree and proves the required ball-decay estimate. The Mass Distribution Principle then converts that estimate into the lower Hausdorff-dimension bound used in the pressure theorem and no-drop argument.

Citation location: Focal Lemma 4.8 and Lemma 4.9, PDF pp. 27–30; source Annals p. 975

Theorem 1.2The pressure formula for the lower and upper endpoint dimensions, the no-dimension-drop identity for the singular intersection, and the criterion for invariance under constant rescaling.

A variational principle in the parametric geometry of numbers→Twisted Diophantine Approximation II: Uniform TheoryHeadline lineageSelf-contained endpoint

The focal sharpness proof constructs an explicit template with prescribed active-time density and invokes the realization theorem to obtain a matrix whose minima track it within bounded error. The resulting pressure and escape proportions give equality in the improved bound.

Citation location: Focal Definition 9.6, Theorem 9.7, and proof of Proposition 2.5, PDF pp. 65–67; source Definition 4.1 and Theorem 4.2

Propositions 2.4–2.5 and 2.7The improved critical escape-of-mass bound, its sharpness for every escape proportion, and the all-power upper bound from rank counts.

Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory→Twisted Diophantine Approximation II: Uniform TheoryHeadline lineageSelf-contained endpoint

Once the finite measure induced on a positive-measure invariant set is invariant under the dense subgroup $\Gamma_A$ , closedness of its stabilizer extends invariance to $H_A$ . Uniqueness of Haar measure then yields the locally full alternative.

Citation location: Focal Lemma 7.1 proof, PDF pp. 48–49; KSS Lemma 1.1.3

Theorem 1.9 and Propositions 1.10–1.11The Hausdorff-measure zero-full law, exact amplification, subgroup structure, reversed endpoint representations, and continuity of endpoint dimensions.

Introduction to Smooth Manifolds→Twisted Diophantine Approximation II: Uniform TheoryHeadline lineageSource-chain endpoint

The Closed Subgroup Theorem supplies the compact Lie-group structure of $H_A$ used to compare Hausdorff and Haar measures and to formulate local fullness in the correct intrinsic dimension.

Citation location: Focal discussion before Theorem 1.9, PDF p. 9; Lee Theorem 20.12

Theorem 1.9 and Propositions 1.10–1.11The Hausdorff-measure zero-full law, exact amplification, subgroup structure, reversed endpoint representations, and continuity of endpoint dimensions.

Measure Theory→Twisted Diophantine Approximation II: Uniform TheoryHeadline lineageSource-chain endpoint

For every nonzero integer vector $\mathbf p_k$ , the character $\mathbf b\mapsto\mathbf p_k\cdot\mathbf b$ is onto the circle. The Haar pushforward theorem therefore gives the exact $2d\delta$ measure of each strip in the focal liminf estimate.

Citation location: Focal Lemma 10.1, PDF p. 69; Halmos §63, Theorem C

Propositions 2.8–2.10The transference nullity and full-space criteria and the exact characterization of trivial shifts, including restricted positive denominators.

Metric theory of inhomogeneous Diophantine approximations with a fixed matrix→Twisted Diophantine Approximation II: Uniform TheoryHeadline lineageSource link checked

The focal proof imports Lemma I with the constant $\kappa_d=2^{1-d}(d!)^2$ . Applied at dual scales, it supplies a denominator for every shift and proves the full-space conclusion; the positive-denominator extension is then obtained internally by adding a fixed positive vector.

Citation location: Focal Lemma 10.2 and proof of Proposition 2.9, PDF pp. 70–71; MN25 Lemma I

Propositions 2.8–2.10The transference nullity and full-space criteria and the exact characterization of trivial shifts, including restricted positive denominators.

On exponents of homogeneous and inhomogeneous Diophantine approximation→Metric theory of inhomogeneous Diophantine approximations with a fixed matrixHeadline lineageSource link partially checked

Moshchevitin--Neckrasov explicitly recall Bugeaud--Laurent Lemma 3 as Lemma I. The dimensions, norms, and constant agree with the normalization imported by the focal paper.

Citation location: MN25 transference section, Lemma I; Bugeaud--Laurent §3, Lemma 3Verification note: The proof-bearing arXiv source was checked; the exact Moscow Mathematical Journal version of record is access-gated.

Lemma IThe finite-scale inhomogeneous transference lemma, in the normalization used to prove the focal full-space criterion.

An Introduction to Diophantine Approximation→On exponents of homogeneous and inhomogeneous Diophantine approximationHeadline lineageSource-chain endpoint

Bugeaud--Laurent reproduce the scale checks for Lemma 3 but invoke Part B of Cassels' Theorem XVII for the decisive inhomogeneous existence assertion.

Citation location: Bugeaud--Laurent §3, Lemma 3 proof; Cassels Chapter V, Theorem XVII, Part B

Lemma 3The proof-bearing formulation from which Moshchevitin--Neckrasov recall Lemma I.