Proof-critical dependence graph
On Some Properties of Irrational Subspaces
A statement-restricted graph for the winning, hyperplane-absolute-winning, exponent-bound, and best-approximation claims in the published paper. Historical and metric-context citations that do not enter those proofs are excluded.
Graph scope11 nodes13 proof-critical linksChecked August 23, 2026
Oriented proof graph
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Proof-critical links
The focal proof iterates Schmidt's escaping lemma on the algebraic bad sets and then uses the permanence and full-dimension properties of winning sets.
Citation location: Focal §1.3 and proof of Theorem 2.1- Theorem 2.1 and Corollary 2.2Winning completely irrational subspaces and the existence/full-dimension conclusion after intersection with badly approximable systems.
The set of badly approximable vectors is strongly incompressible→On Some Properties of Irrational SubspacesHeadline lineageSelf-contained leaf
The focal HAW argument uses the game's countable-intersection property and its relation to the original Schmidt game exactly as established by Broderick–Fishman–Kleinbock–Reich–Weiss.
Citation location: Focal §1.2 and proof of Theorem 2.3- Theorem 2.3Hyperplane-absolute-winning versions of the completely irrational and badly approximable completely irrational sets.
Badly approximable systems of linear forms→On Some Properties of Irrational SubspacesHeadline lineageVerified
Schmidt's winning theorem for badly approximable systems is intersected with the focal paper's winning set of completely irrational subspaces.
Citation location: Focal Proposition 1.8 and Corollary after Theorem 2.1- Theorem 2.1 and Corollary 2.2Winning completely irrational subspaces and the existence/full-dimension conclusion after intersection with badly approximable systems.
Badly approximable systems of affine forms and incompressibility on fractals→On Some Properties of Irrational SubspacesHeadline lineageVerified
The hyperplane-absolute-winning theorem for badly approximable systems is intersected with the focal paper's HAW completely irrational set.
Citation location: Focal Proposition 1.8 and proof of Theorem 2.3- Theorem 2.3Hyperplane-absolute-winning versions of the completely irrational and badly approximable completely irrational sets.
Singular vectors on manifolds and fractals→On Some Properties of Irrational SubspacesHeadline lineageVerified
The focal paper imports the ordinary-exponent upper bound for vectors in a badly approximable subspace, then combines it with the dimension-sensitive ratio inequality.
Citation location: Focal Proposition 3.1 and proof of Theorem 3.4- Theorem 3.4The uniform-exponent upper bound for every vector in a badly approximable completely irrational subspace.
An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation→On Some Properties of Irrational SubspacesHeadline lineageVerified
The sharp ordinary/uniform exponent ratio is the quantitative inequality from which the focal polynomial bound is derived.
Citation location: Focal Proposition 3.2 and discussion before Theorem 3.4- Theorem 3.4The uniform-exponent upper bound for every vector in a badly approximable completely irrational subspace.
Applications of Siegel's Lemma to a system of linear forms and its minimal points→On Some Properties of Irrational SubspacesHeadline lineageVerified
The arXiv source supplies the version of the ratio estimate indexed by rational dimension, allowing the focal argument to treat every vector in the subspace.
Citation location: Focal Proposition 3.3 and proof of Theorem 3.4- Theorem 3.4The uniform-exponent upper bound for every vector in a badly approximable completely irrational subspace.
Khintchine's singular Diophantine systems and their applications→On Some Properties of Irrational SubspacesHeadline lineageSelf-contained leaf
The survey's Corollary 4 to Theorem 7 gives the two-or-four alternative; complete irrationality excludes the two-dimensional branch.
Citation location: Focal final subsection of §3- Corollary 3.7For a good completely irrational two-dimensional subspace of , the eventual best approximations span all four dimensions.
On badly approximable numbers and certain games→Badly approximable systems of linear formsHeadline lineageVerified
The 1969 construction uses the game and permanence machinery introduced in Schmidt's 1966 paper to establish the winning result for systems of linear forms.
Citation location: Schmidt 1969 game construction; Schmidt 1966- Winning theorem for badly approximable systemsShows that the parameter set of badly approximable systems of linear forms is winning.
Badly approximable systems of linear forms→Badly approximable systems of affine forms and incompressibility on fractalsHeadline lineageVerified
The proof of the HAW theorem restates Schmidt's Lemmas 1 and 2 as Propositions 5.1 and 5.2. Their dimension bounds for the relevant integral solution spaces are the input used to place every dangerous family inside a single deletable hyperplane.
Citation location: Broderick–Fishman–Simmons §5, Propositions 5.1–5.2, and §6; Schmidt 1969, Lemmas 1–2- HAW theorem for badly approximable systemsUpgrades the relevant set of systems of linear forms to hyperplane absolute winning.
Diophantine Approximation and Diophantine Equations→An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximationHeadline lineageTerminal source
Schmidt's inequality on heights is applied inductively to selected rational subspaces throughout the proof of the optimal ratio theorem.
Citation location: Marnat–Moshchevitin §4 and proof of the main theorem- Main ratio theoremGives the sharp lower bound for the ratio of ordinary to uniform Diophantine exponents.
An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation→Singular vectors on manifolds and fractalsHeadline lineageVerified
The second part of Proposition 5.1 invokes the optimal ordinary/uniform ratio theorem to convert the ordinary-exponent bound into a uniform-exponent bound.
Citation location: Kleinbock–Moshchevitin–Weiss arXiv:1912.13070v2, §5, Proposition 5.1- Proposition 5.1Bounds the ordinary exponent of vectors lying in a badly approximable subspace and combines it with the uniform/ordinary ratio estimate.
An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation→Applications of Siegel's Lemma to a system of linear forms and its minimal pointsHeadline lineageVerified
Schleischitz records the Marnat–Moshchevitin theorem as Theorem 2.1 and then formulates the rational-dimension consequence used by the focal paper in Theorem 2.4, equation (6).
Citation location: Schleischitz arXiv:1904.06121v4, §2.1, Theorems 2.1 and 2.4- Theorem 2.4, equation (6), rational-dimension specializationRecords the ratio bound in the dimension-sensitive form used for vectors that are not necessarily totally irrational.