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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Observed surface forms (4)

Surface form	Occurrences
Roth's theorem	0
Roth's theorem on Diophantine approximation	2
Roth's theorem on three-term arithmetic progressions	1
Thue–Siegel–Roth theorem	1

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 ⓘ

Referenced by (5)

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

Diophantine approximation → hasKeyResult → Roth theorem ⓘ

Klaus Roth → knownFor → Roth theorem ⓘ

this entity surface form: Roth's theorem on Diophantine approximation

Klaus Roth → notableWork → Roth theorem ⓘ

this entity surface form: Roth's theorem on Diophantine approximation

Klaus Roth → notableWork → Roth theorem ⓘ

this entity surface form: Roth's theorem on three-term arithmetic progressions

Carl Ludwig Siegel → notableWork → Roth theorem ⓘ

this entity surface form: Thue–Siegel–Roth theorem