Proof-critical dependence graph
Uniform Diophantine approximation with restrictions via total density of collections of subspaces
A statement-restricted graph for the total-density criteria, their uniform-approximation consequences, the convex-hull criterion, and the separately marked optimality branch. Context, historical attributions, alternative proof sources, and arguments reproduced inside the focal paper are excluded.
Graph scope4 nodes3 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
- Dashed square: a terminal book
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Every visible arrow is documented in the evidence ledger below. A non-book leaf means that no earlier proof-critical source was identified for the marked statement—not that the paper has no other citations.
Evidence ledger
Proof-critical links
Singularity, weighted uniform approximation, intersections and rates→Uniform Diophantine approximation with restrictions via total density of collections of subspacesHeadline lineageSelf-contained leaf
After the focal paper proves total density or finds a totally dense subcollection, KMWW Theorem 1.5 is the imported mechanism that produces uncountably many dense uniformly approximable points while avoiding the prescribed analytic exceptional sets.
Citation location: Focal Theorem B / simplified KMWW Theorem 1.5, PDF p. 3; applications throughout §1.3 and proof of Theorem 1.5, PDF p. 35; focal bibliography [KMWW] cites the Compositio article.- Theorem 1.5 and its uniform-approximation conclusionEstablishes total density for the two principal restricted-approximation families and deduces uncountable dense sets avoiding prescribed analytic submanifolds.
Convexity→Uniform Diophantine approximation with restrictions via total density of collections of subspacesHeadline lineageTerminal source
The focal generalized Carathéodory lemma starts from the classical finite convex-combination theorem, then adds its own partition-sensitive argument; that modification is used in the proof of the convex-hull criterion.
Citation location: Focal PDF p. 24, classical Carathéodory theorem immediately before the generalized lemma; citation ‘see [Eggleston, Chapter 2.2] for details’.- Theorem 2.3Gives the convex-hull criterion for property (TDS), using the focal generalized Carathéodory lemma.
Convex Optimization→Uniform Diophantine approximation with restrictions via total density of collections of subspacesAncillary claimTerminal source
The separating-hyperplane theorem provides a functional strictly positive on the relevant direction set, which yields a nonempty open region missed by the subspace collection and underpins the focal optimality construction.
Citation location: Focal PDF p. 19 and the later optimality proof; citation [BV04, Chapter 2.5.1].- Product-density characterization and optimality propositionsThe negative direction separates the origin from the relevant convex hull to construct an open region missed by the subspace family.