Proof-critical dependence graph
Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle
A statement-restricted graph for the zero–full measure theorem, its weighted transference inputs, and the separately marked second-moment and geometry-of-numbers branches. Contextual citations, analogies, alternative routes, and sources whose needed result is reproved in the focal paper are excluded.
Graph scope18 nodes19 proof-critical linksChecked August 23, 2026
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Proof-critical links
Measure theoretic laws for limsup sets defined by rectangles→Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principleHeadline lineageVerified
The focal proof invokes Kleinbock–Wang Theorem 2.7, restated as Theorem B, for both convergence and divergence after proving the internal equivalence of series.
Citation location: Focal Theorem B, PDF p. 5; proof of Theorem 1.3, PDF pp. 24–26; bibliography [KW23] cites Adv. Math. 428 (2023).- Theorem 1.3Zero–full Lebesgue-measure law for weighted uniform inhomogeneous approximation.
A strengthening of Mahler’s transference theorem→Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principleHeadline lineageVerified
Focal Lemma D(1) imports German–Evdokimov Theorem 1, while Lemma D(2) invokes their recalled Theorem D; these estimates power the weighted homogeneous/inhomogeneous transference arguments.
Citation location: Focal Lemma D, PDF pp. 11–12, citation [GK15, Theorem 1 and Theorem D].- Lemma D and weighted transference theoremsQuantitative pseudo-compound-body transference used to pass between homogeneous and inhomogeneous approximation statements.
Metric Theory of Diophantine Approximations→Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principleHeadline lineageTerminal source
The focal second-moment calculation ends by applying Sprindzhuk's limsup inequality to obtain the required positive/full measure conclusion.
Citation location: Focal proof around PDF p. 21: ‘Lemma 5, §3, Ch. 1 in [Spr]’.- Theorem 1.5Measure conclusion obtained from a sharp second-moment limsup estimate.
An Introduction to the Geometry of Numbers→Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principleAncillary claimTerminal source
Cassels is the focal paper's reference for dual-lattice and classical geometry-of-numbers background surrounding its Minkowski statement.
Citation location: Focal geometry-of-numbers setup, PDF pp. 10–13, citation [cas].- Theorem 3.1 and dual-body setupClassical geometry-of-numbers input used in the paper's lattice formulation.
On the application of the Borel–Cantelli lemma→Measure theoretic laws for limsup sets defined by rectanglesHeadline lineageSelf-contained leaf
The second-moment lower bound is applied locally to rectangular events; the resulting positive limsup measure is upgraded to full measure after the density step.
Citation location: Chung–Erdős pp. 180–181, equations (4)–(6); Kleinbock–Wang Lemma 3.1 and Theorem 2.5 Steps 2–4, pp. 7–12.- Theorem 2.5Abstract rectangular ubiquity theorem used in the divergence proof of Theorem 2.7.
Measure theoretic laws for lim sup sets→Measure theoretic laws for limsup sets defined by rectanglesHeadline lineageVerified
BDV Proposition 1 supplies the local positive-density-to-full-measure principle reproduced as Kleinbock–Wang Lemma 3.2 and applied at the end of Theorem 2.5.
Citation location: BDV Proposition 1, printed p. 29; Kleinbock–Wang Lemma 3.2, p. 7, and proof p. 12.- Theorem 2.5Abstract rectangular ubiquity theorem used in the divergence proof of Theorem 2.7.
Lectures on Analysis on Metric Spaces→Measure theoretic laws for limsup sets defined by rectanglesHeadline lineageTerminal source
The metric 5r covering theorem supplies the selection idea for the aligned-rectangle covering lemma used in the first two steps of Theorem 2.5.
Citation location: Heinonen Theorem 1.2; Kleinbock–Wang Lemma 3.3 and Theorem 2.5 Steps 1–2, pp. 7–9.- Theorem 2.5Abstract rectangular ubiquity theorem used in the divergence proof of Theorem 2.7.
Geometrie der Zahlen→Measure theoretic laws for limsup sets defined by rectanglesHeadline lineageTerminal source
Minkowski's convex-body theorem gives the nonzero integer vector satisfying the simultaneous rectangular inequalities, establishing the ubiquity input and divergence half of Theorem 2.7.
Citation location: Kleinbock–Wang p. 18 and Lemma 6.2 proof, p. 19.- Lemma 6.2Minkowski-based ubiquity input for the rectangular Khintchine–Groshev theorem.
- Theorem 2.7General Khintchine–Groshev zero–full law for rectangular approximation used as Theorem B by the focal paper.
Federer's adequate-family selection theorem and inner regularity provide the disjoint covering and closed approximation used in BDV Lemmas 6–7, which prove Proposition 1.
Citation location: Federer §§2.8.7 and 2.2.2; BDV printed pp. 26–29.- Proposition 1Positive local density implies full measure.
Ein Übertragungsprinzip für konvexe Körper→A strengthening of Mahler’s transference theoremHeadline lineageVerified
German–Evdokimov explicitly restate Mahler's consecutive-minima transference theorem as Theorem D; the focal paper uses that restatement in Lemma D(2).
Citation location: German–Evdokimov printed p. 63, Theorem D, citation [2]; Mahler pp. 93–102.- Theorem DMahler's consecutive-minima transference inequality used by the focal proof of Lemma D(2).
On Diophantine exponents and Khintchine’s transference principle→A strengthening of Mahler’s transference theoremHeadline lineageVerified
The section-dual covariance imported as German–Evdokimov Lemma 1(2) is needed for Corollary 1, which combines with Lemma 2 in the proof of Theorem 1.
Citation location: German–Evdokimov Lemma 1(2), printed p. 66, citation [5]; Theorem 1 proof §6, p. 68.- Theorem 1Strengthened consecutive-minima transference estimate imported in focal Lemma D(1).
A geometric inequality with applications to linear forms→A strengthening of Mahler’s transference theoremHeadline lineageVerified
Vaaler's cube-section lower bound yields German–Evdokimov Lemma 3 and Corollary 1, which are used with Lemma 2 to prove Theorem 1.
Citation location: German–Evdokimov §5, printed pp. 67–68, citation [10]; Vaaler corollary, p. 544.- Theorem 1Strengthened consecutive-minima transference estimate imported in focal Lemma D(1).
Geometrie der Zahlen→A strengthening of Mahler’s transference theoremHeadline lineageTerminal source
Minkowski's theorem on successive minima is the external inequality used in German–Evdokimov Lemma 2, one of the two immediate inputs to the proof of Theorem 1.
Citation location: German–Evdokimov Lemma 2 proof, printed pp. 66–67, and Theorem 1 proof §6.- Theorem 1Strengthened consecutive-minima transference estimate imported in focal Lemma D(1).
Mahler's consecutive-minima transference argument rests on the classical geometry-of-numbers minima inequalities; expansion stops at Minkowski's published book.
Citation location: Mahler 1939 proof; German–Evdokimov discussion surrounding equations (10)–(11), p. 63.- Consecutive-minima transference theoremRelates successive minima of a parallelepiped and its pseudo-compound dual.
Diophantine Approximation→On Diophantine exponents and Khintchine’s transference principleHeadline lineageTerminal source
Schmidt's exterior-algebra theorem supplies German's Proposition 3; German then applies it in the proof of Lemma 3 to obtain the cofactor transformation rule needed by German–Evdokimov.
Citation location: German 2012 Proposition 3 and Lemma 3, printed pp. 35–38; citation [8, Chapter VII §3, Theorem 1].- Proposition 3 and Lemma 3Cofactor/section-dual covariance imported as German–Evdokimov Lemma 1(2).
Unimodality and dominance for symmetric random vectors→A geometric inequality with applications to linear formsHeadline lineageVerified
Vaaler's Lemma 4 explicitly invokes Kanter Corollary 3.2 to preserve peakedness under products; that comparison drives the proof of Theorem 1 and its cube-section corollary.
Citation location: Vaaler Lemma 4, printed p. 546; Kanter Corollary 3.2, printed p. 76.- Corollary to Theorem 1Every central hyperplane section of the cube has the lower-volume bound used in German–Evdokimov Corollary 1.
On logarithmic concave measures and functions→Unimodality and dominance for symmetric random vectorsHeadline lineageVerified
Kanter's log-concavity step uses Prékopa's theorem that integration over variables preserves logarithmic concavity, supporting the product/convolution peakedness result behind Corollary 3.2.
Citation location: Kanter log-concavity lemmas preceding Corollary 3.2; Prékopa Theorem 6, p. 342.- Corollary 3.2Peakedness is preserved under products, the key input in Vaaler's cube-versus-Gaussian comparison.
On a certain converse of Hölder’s inequality. II→On logarithmic concave measures and functionsHeadline lineageVerified
Prékopa derives the multivariable marginal theorem from the one-dimensional integral inequality proved by Leindler.
Citation location: Prékopa proof chain to Theorem 6; Leindler equation (4) and theorem, pp. 217–223.- Theorem 6Marginalization preserves log-concavity, as used in Kanter's proof.
Logarithmic concave measures with application to stochastic programming→On a certain converse of Hölder’s inequality. IIHeadline lineageSelf-contained leaf
Leindler explicitly follows Prékopa's approximation argument to pass from step functions to arbitrary measurable functions; that is the only inherited step needed for the marked inequality.
Citation location: Leindler proof after the step-function case; Prékopa 1971 proof of Theorem 1.- Integral converse-to-Hölder inequalityOne-dimensional integral inequality imported in Prékopa's log-concavity proof.