Subspace theorem

E637309

result in Diophantine approximation theorem in number theory

The Subspace theorem is a fundamental result in Diophantine approximation that describes how solutions to certain inequalities involving linear forms over algebraic numbers must lie in a finite union of proper subspaces.

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All labels observed (3)

Label	Occurrences
Subspace theorem canonical	2
absolute Subspace theorem	1
quantitative Subspace theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T7030804 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Subspace theorem
Context triple: [Diophantine approximation, hasKeyResult, Subspace theorem]

A. Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
B. Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
C. Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
D. Bernstein theorem
Bernstein theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
E. Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Subspace theorem
Target entity description: The Subspace theorem is a fundamental result in Diophantine approximation that describes how solutions to certain inequalities involving linear forms over algebraic numbers must lie in a finite union of proper subspaces.

A. Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
B. Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
C. Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
D. Bernstein theorem
Bernstein theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
E. Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	result in Diophantine approximation ⓘ theorem in number theory ⓘ
alsoKnownAs	Schmidt Subspace theorem ⓘ linked to: Schmidt subspace theorem
appliesTo	Diophantine inequalities ⓘ S-unit equations ⓘ systems of linear forms ⓘ
assumption	coefficients lie in a number field ⓘ linear forms are linearly independent ⓘ solutions considered in integer or S-integer points ⓘ
centralConcept	proper linear subspaces of affine or projective space ⓘ
concerns	approximations to algebraic numbers ⓘ linear forms in several variables ⓘ
conclusion	outside a finite union of proper subspaces only finitely many solutions exist ⓘ solutions lie in finitely many proper subspaces ⓘ
describes	structure of solutions to certain Diophantine inequalities ⓘ
domain	algebraic number fields ⓘ
field	Diophantine approximation ⓘ number theory ⓘ
generalizes	Thue–Siegel–Roth theorem ⓘ
hasVariant	Evertse–Schlickewei–Schmidt quantitative version ⓘ Schlickewei’s p-adic generalization ⓘ absolute Subspace theorem ⓘ linked to: Subspace theorem p-adic Subspace theorem ⓘ linked to: Schmidt subspace theorem quantitative Subspace theorem ⓘ linked to: Subspace theorem
implies	Roth’s theorem on Diophantine approximation ⓘ linked to: Thue–Siegel–Roth theorem
inspired	applications to transcendence theory ⓘ developments in higher-dimensional Diophantine approximation ⓘ
involves	Archimedean and non-Archimedean valuations ⓘ inequalities with respect to a finite set of places ⓘ product of absolute values of linear forms ⓘ
namedAfter	Wolfgang M. Schmidt ⓘ
originallyProvedBy	Wolfgang M. Schmidt ⓘ
relatedTo	Diophantine geometry ⓘ Vojta’s conjectures ⓘ geometry of numbers ⓘ height functions on algebraic numbers ⓘ
strengthenedBy	Evertse–Schlickewei–Schmidt theorem ⓘ Evertse’s quantitative refinements ⓘ Schlickewei’s quantitative refinements ⓘ
type	finiteness theorem ⓘ subspace-type Diophantine approximation theorem ⓘ
typicalSetting	solutions in projective n-space over a number field ⓘ
usedFor	finiteness results for integral points on varieties ⓘ results on S-unit equations ⓘ results on exponential Diophantine equations ⓘ results on linear recurrence sequences ⓘ
yearProved	1972 ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Diophantine approximation → hasKeyResult → Subspace theorem ⓘ

Subspace theorem → hasVariant → absolute Subspace theorem ⓘ

linked to: Subspace theorem

Subspace theorem → hasVariant → quantitative Subspace theorem ⓘ

linked to: Subspace theorem

Siegel’s lemma → relatedTo → Subspace theorem ⓘ

All labels observed (3)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (4)

How this entity was disambiguated

How these facts were elicited