Roth theorem

E637307

result in Diophantine approximation theorem in number theory

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.

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All labels observed (6)

Label	Occurrences
Roth's theorem	5
Roth's theorem on Diophantine approximation	2
Roth theorem canonical	1
Roth's theorem on three-term arithmetic progressions	1
Roth’s theorem	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 ⓘ
appliesTo	irrational algebraic numbers of any degree at least 2 ⓘ
cannotBeImprovedTo	exponent greater than 2 for all algebraic irrationals ⓘ
comparesWith	Dirichlet's approximation theorem ⓘ linked to: Dirichlet approximation theorem
concerns	algebraic irrational numbers ⓘ approximation of algebraic numbers by rationals ⓘ rational approximations ⓘ
context	Diophantine approximation on the real line ⓘ
contrastsWith	Liouville numbers ⓘ
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 ⓘ 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 ⓘ linked to: Siegel's theorem on integral points
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 ⓘ linked to: Klaus Roth
provedBy	Klaus Friedrich Roth ⓘ linked to: Klaus Roth
publishedIn	1955 ⓘ
refines	Dirichlet-type bounds for algebraic irrationals ⓘ
relatedTo	Schmidt subspace theorem ⓘ Thue–Siegel–Roth theorem ⓘ
sharpens	Liouville's theorem on Diophantine approximation ⓘ linked to: Liouville's inequality in Diophantine approximation
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

Referenced by (11)

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

Diophantine approximation → hasKeyResult → Roth theorem ⓘ

Klaus Roth → knownFor → Roth's theorem on Diophantine approximation ⓘ

linked to: Roth theorem

Klaus Roth → notableWork → Roth's theorem on Diophantine approximation ⓘ

linked to: Roth theorem

Klaus Roth → notableWork → Roth's theorem on three-term arithmetic progressions ⓘ

linked to: Roth theorem

Carl Ludwig Siegel → notableWork → Thue–Siegel–Roth theorem ⓘ

linked to: Roth theorem

abc conjecture → relatedTo → Roth’s theorem ⓘ

linked to: Roth theorem

Dirichlet approximation theorem → relatedTo → Roth's theorem ⓘ

linked to: Roth theorem

Siegel's theorem on integral points → relatedTo → Roth's theorem ⓘ

linked to: Roth theorem

Siegel's theorem on integral points → strengthenedBy → Roth's theorem ⓘ

linked to: Roth theorem

Liouville's inequality in Diophantine approximation → relatedConcept → Roth's theorem ⓘ

linked to: Roth theorem

Liouville's inequality in Diophantine approximation → isWeakerThan → Roth's theorem ⓘ

linked to: Roth theorem

All labels observed (6)

How this entity was disambiguated Show

Statements (48)

How these facts were elicited Show

Referenced by (11)

How this entity was disambiguated

How these facts were elicited