Proof-critical dependence graph

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.

Graph scope7 nodes6 proof-critical linksChecked August 23, 2026

Oriented proof graph

Dependence map

Arrows point from a prerequisite toward the paper whose marked statement uses it.

Solid arrow: headline proof lineage
Dashed arrow: a separately marked side or appendix claim
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Evidence ledger

Proof-critical links

On exponents of homogeneous and inhomogeneous Diophantine approximation→Metric theory of inhomogeneous Diophantine approximations with a fixed matrixHeadline lineagePartially verified

Two distinct imported results are proof-critical: Bugeaud–Laurent Lemma 3 is recalled as Lemma I and drives the uniform/asymptotic transference arguments, including Lemma 7.9; Lemma 1 supplies Proposition 5.1(a), the geometric growth used to extract lacunary sequences and to make several convergence and construction arguments work.

Citation location: Focal Proposition 5.1(a), p. 23, citation [10]; focal Lemma I, p. 28, citation [10]; source §2 Lemma 1 and §3 Lemma 3Verification note: Statements and proofs were primary-checked in arXiv:math/0406065; the exact cited journal VOR is access-gated.

Theorems 1.3(a) and 1.4(a)The full-measure uniform and asymptotic inhomogeneous conclusions obtained from the homogeneous lower bound by the recalled transference lemma.
Theorems 3.2 and 3.4The two hyperplane-absolute-winning conclusions on affine subspaces. Their proof combines geometric growth of best-approximation denominators, the lacunary escaping-set theorem, and closure of HAW under finite/countable intersections.
Theorem 3.3The almost-everywhere asymptotic inhomogeneous approximation conclusion. Lemma 7.9 combines the transference input with the second-moment limsup inequality.
Theorem 3.6The convergence/nullity result whose summability mechanism uses the imported geometric growth of best-approximation denominators.
Theorem 3.8(b)The uncountability construction for prescribed inhomogeneous approximation, where Proposition 5.1 supplies the geometric best-approximation growth used in Theorem 9.1(b).

Badly approximable systems of affine forms and incompressibility on fractals→Metric theory of inhomogeneous Diophantine approximations with a fixed matrixHeadline lineagePartially verified

The focal Proposition 7.7 is a direct special case of the lacunary-matrix escaping theorem. It is applied to projected best-approximation vectors in Lemma 7.8 to obtain the HAW subset used in Theorem 7.2 and hence Theorems 3.2 and 3.4.

Citation location: Focal Proposition 7.7, p. 33, citation [13, Theorem 4.1]; source §4, Theorem 4.1 and proofVerification note: The complete arXiv source was checked; the exact cited Journal of Number Theory VOR is access-gated.

Theorems 3.2 and 3.4The two hyperplane-absolute-winning conclusions on affine subspaces. Their proof combines geometric growth of best-approximation denominators, the lacunary escaping-set theorem, and closure of HAW under finite/countable intersections.

The set of badly approximable vectors is strongly $C^1$ incompressible→Metric theory of inhomogeneous Diophantine approximations with a fixed matrixHeadline lineageVerified

The countable-intersection property for HAW sets combines the finitely many lacunary escaping sets in Lemma 7.8; the recalled HAW-to-Schmidt-winning/full-dimension implication supports the advertised strength of the conclusions.

Citation location: Focal HAW-property list, p. 10, citation [12], and Lemma 7.8, pp. 33–34; source Proposition 2.3(a)–(b), printed p. 323

Theorems 3.2 and 3.4The two hyperplane-absolute-winning conclusions on affine subspaces. Their proof combines geometric growth of best-approximation denominators, the lacunary escaping-set theorem, and closure of HAW under finite/countable intersections.

Metric Theory of Diophantine Approximations→Metric theory of inhomogeneous Diophantine approximations with a fixed matrixHeadline lineageTerminal source

The focal paper explicitly recalls the second-moment limsup bound in Sprindžuk's exact form as Lemma J. It closes the proof of Theorem 3.1 and, together with Lemma I, proves Lemma 7.9 and Theorem 3.3.

Citation location: Focal Lemma J, p. 28, citation [49, Chapter 1, §3, Lemma 5]; applications p. 36 and p. 37

Theorem 3.1The nullity statement for uniformly improvable shifts on a nonexceptional affine subspace; the proof closes with the recalled second-moment limsup inequality.
Theorem 3.3The almost-everywhere asymptotic inhomogeneous approximation conclusion. Lemma 7.9 combines the transference input with the second-moment limsup inequality.

Fractal Geometry: Mathematical Foundations and Applications→Metric theory of inhomogeneous Diophantine approximations with a fixed matrixAncillary claimTerminal source

Falconer's product formula identifies the Hausdorff dimension of the incidence set $E\times Gr(l,r)$ in Lemma 6.2. Projection then gives the exceptional-subspace bound stated in Corollary 6.3.

Citation location: Focal Lemma 6.2 proof, p. 26, citation [19, Corollary 7.4]

Corollary 6.3The Hausdorff-dimension and measure-zero estimate for exceptional Grassmannian subspaces. This is a separately marked ancillary branch.

An Introduction to Diophantine Approximation→On exponents of homogeneous and inhomogeneous Diophantine approximationHeadline lineageTerminal source

Bugeaud–Laurent reproduce their Lemma 3 proof, but the decisive existence assertion is explicitly Part B of Cassels' Theorem XVII. The remainder of their proof checks Cassels' hypothesis under the stated homogeneous lower bound.

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

Lemma 3Turns a uniform lower bound for the transpose into an inhomogeneous approximation for every shift. Its reproduced proof invokes Cassels, Chapter V, Theorem XVII, Part B.