Subspace theorem
E637309
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Subspace theorem canonical | 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.
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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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| hasVariant |
Evertse–Schlickewei–Schmidt quantitative version
ⓘ
Schlickewei’s p-adic generalization NERFINISHED ⓘ absolute Subspace theorem NERFINISHED ⓘ p-adic Subspace theorem NERFINISHED ⓘ quantitative Subspace theorem NERFINISHED ⓘ |
| implies | Roth’s theorem on Diophantine approximation NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| originallyProvedBy | Wolfgang M. Schmidt NERFINISHED ⓘ |
| relatedTo |
Diophantine geometry
NERFINISHED
ⓘ
Vojta’s conjectures NERFINISHED ⓘ geometry of numbers ⓘ height functions on algebraic numbers ⓘ |
| strengthenedBy |
Evertse–Schlickewei–Schmidt theorem
NERFINISHED
ⓘ
Evertse’s quantitative refinements ⓘ Schlickewei’s quantitative refinements ⓘ |
| type |
finiteness theorem
ⓘ
subspace-type Diophantine approximation theorem NERFINISHED ⓘ |
| 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
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Instruction
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Input
Subject: Subspace theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.