Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle

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.

• Theorem 1.3Zero–full Lebesgue-measure law for weighted uniform inhomogeneous approximation.

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.

• Lemma D and weighted transference theoremsQuantitative pseudo-compound-body transference used to pass between homogeneous and inhomogeneous approximation statements.

The focal second-moment calculation ends by applying Sprindzhuk's limsup inequality to obtain the required positive/full measure conclusion.

• Theorem 1.5Measure conclusion obtained from a sharp second-moment limsup estimate.

Cassels is the focal paper's reference for dual-lattice and classical geometry-of-numbers background surrounding its Minkowski statement.

• Theorem 3.1 and dual-body setupClassical geometry-of-numbers input used in the paper's lattice formulation.

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.

• Theorem 2.5Abstract rectangular ubiquity theorem used in the divergence proof of Theorem 2.7.

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.

• Theorem 2.5Abstract rectangular ubiquity theorem used in the divergence proof of Theorem 2.7.

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.

• Theorem 2.5Abstract rectangular ubiquity theorem used in the divergence proof of Theorem 2.7.

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.

• 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.

• Proposition 1Positive local density implies full measure.

German–Evdokimov explicitly restate Mahler's consecutive-minima transference theorem as Theorem D; the focal paper uses that restatement in Lemma D(2).

• Theorem DMahler's consecutive-minima transference inequality used by the focal proof of Lemma D(2).

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.

• Theorem 1Strengthened consecutive-minima transference estimate imported in focal Lemma D(1).

Vaaler's cube-section lower bound yields German–Evdokimov Lemma 3 and Corollary 1, which are used with Lemma 2 to prove Theorem 1.

• Theorem 1Strengthened consecutive-minima transference estimate imported in focal Lemma D(1).

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.

• 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.

• Consecutive-minima transference theoremRelates successive minima of a parallelepiped and its pseudo-compound dual.

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.

• Proposition 3 and Lemma 3Cofactor/section-dual covariance imported as German–Evdokimov Lemma 1(2).

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.

• Corollary to Theorem 1Every central hyperplane section of the cube has the lower-volume bound used in German–Evdokimov Corollary 1.

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.

• Corollary 3.2Peakedness is preserved under products, the key input in Vaaler's cube-versus-Gaussian comparison.

Prékopa derives the multivariable marginal theorem from the one-dimensional integral inequality proved by Leindler.

• Theorem 6Marginalization preserves log-concavity, as used in Kanter's proof.

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.

• Integral converse-to-Hölder inequalityOne-dimensional integral inequality imported in Prékopa's log-concavity proof.