Roth theorem
E637307
Roth's theorem is a fundamental result in Diophantine approximation that gives an essentially optimal bound on how well algebraic irrational numbers can be approximated by rational numbers.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Roth's theorem on Diophantine approximation | 2 |
| Roth theorem canonical | 1 |
| Roth's theorem on three-term arithmetic progressions | 1 |
| Thue–Siegel–Roth theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7030801 — 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: Roth theorem Context triple: [Diophantine approximation, hasKeyResult, Roth theorem]
-
A.
Green–Tao theorem
The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
-
B.
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.
-
C.
Erdős–Turán conjecture
The Erdős–Turán conjecture is an unsolved problem in additive number theory asserting that any subset of the positive integers with divergent sum of reciprocals must contain arbitrarily long arithmetic progressions.
-
D.
Sylvester–Gallai theorem
The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
-
E.
Chevalley–Warning theorem
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
- 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: Roth theorem Target entity description: Roth's theorem is a fundamental result in Diophantine approximation that gives an essentially optimal bound on how well algebraic irrational numbers can be approximated by rational numbers.
-
A.
Green–Tao theorem
The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
-
B.
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.
-
C.
Erdős–Turán conjecture
The Erdős–Turán conjecture is an unsolved problem in additive number theory asserting that any subset of the positive integers with divergent sum of reciprocals must contain arbitrarily long arithmetic progressions.
-
D.
Sylvester–Gallai theorem
The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
-
E.
Chevalley–Warning theorem
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
result in Diophantine approximation
ⓘ
theorem in number theory ⓘ |
| alsoKnownAs | Thue–Siegel–Roth theorem NERFINISHED ⓘ |
| appliesTo | irrational algebraic numbers of any degree at least 2 ⓘ |
| cannotBeImprovedTo | exponent greater than 2 for all algebraic irrationals ⓘ |
| comparesWith | Dirichlet's approximation theorem NERFINISHED ⓘ |
| concerns |
algebraic irrational numbers
ⓘ
approximation of algebraic numbers by rationals ⓘ rational approximations ⓘ |
| context | Diophantine approximation on the real line ⓘ |
| contrastsWith | Liouville numbers NERFINISHED ⓘ |
| doesNotApplyTo |
rational numbers
ⓘ
transcendental numbers in general ⓘ |
| field |
Diophantine approximation
ⓘ
number theory ⓘ |
| gives | essentially optimal exponent 2 in rational approximation of algebraic irrationals ⓘ |
| givesBoundOn | irrationality measure of algebraic numbers ⓘ |
| hasConsequence |
algebraic irrational numbers are not Liouville numbers
ⓘ
only finitely many very good rational approximations to a given algebraic irrational ⓘ |
| hasGeneralization |
Schmidt subspace theorem
NERFINISHED
ⓘ
higher-dimensional Diophantine approximation results ⓘ |
| implies |
irrational algebraic numbers cannot be approximated too closely by rationals
ⓘ
irrationality measure of any irrational algebraic number is 2 ⓘ |
| improvesOn | Thue–Siegel theorem NERFINISHED ⓘ |
| inspired | work on quantitative versions of Diophantine approximation ⓘ |
| involves |
Diophantine inequalities
ⓘ
degree of algebraic numbers ⓘ height of algebraic numbers ⓘ |
| isCornerstoneOf | modern Diophantine approximation ⓘ |
| isOptimalIn | exponent of q in the approximation inequality ⓘ |
| mathematicalDomain |
algebraic number theory
ⓘ
analytic number theory ⓘ |
| motivated | later developments in Diophantine approximation ⓘ |
| namedAfter | Klaus Friedrich Roth NERFINISHED ⓘ |
| provedBy | Klaus Friedrich Roth NERFINISHED ⓘ |
| publishedIn | 1955 ⓘ |
| refines | Dirichlet-type bounds for algebraic irrationals ⓘ |
| relatedTo |
Schmidt subspace theorem
NERFINISHED
ⓘ
Thue–Siegel–Roth theorem NERFINISHED ⓘ |
| sharpens | Liouville's theorem on Diophantine approximation NERFINISHED ⓘ |
| states | if α is an irrational algebraic number and ε > 0 then |α − p/q| < 1/q^{2+ε} has only finitely many rational solutions p/q ⓘ |
| strengthens | Liouville-type inequalities for algebraic numbers ⓘ |
| typeOfBound | upper bound on quality of rational approximation ⓘ |
| usedIn |
metric Diophantine approximation
ⓘ
results on uniform distribution ⓘ transcendence theory ⓘ |
| usedMethod | Thue–Siegel method with new ideas ⓘ |
| yearProved | 1955 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Roth theorem Description of subject: Roth's theorem is a fundamental result in Diophantine approximation that gives an essentially optimal bound on how well algebraic irrational numbers can be approximated by rational numbers.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Roth's theorem on Diophantine approximation
this entity surface form:
Roth's theorem on Diophantine approximation
this entity surface form:
Roth's theorem on three-term arithmetic progressions
this entity surface form:
Thue–Siegel–Roth theorem