Khintchine theorem

E637305

mathematical theorem result in metric Diophantine approximation

Khintchine theorem is a fundamental result in metric Diophantine approximation that characterizes, via a simple convergence–divergence criterion, when almost all real numbers admit infinitely many rational approximations of a prescribed quality.

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Observed surface forms (2)

Surface form	Occurrences
Khinchin's theorem on continued fractions	1
Khintchine–Groshev theorem	1

Statements (38)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in metric Diophantine approximation ⓘ
alternativeName	Khinchin theorem NERFINISHED ⓘ
appliesTo	simultaneous approximation in higher dimensions via extensions ⓘ
areaOfInfluence	fractal geometry of Diophantine sets ⓘ probabilistic number theory ⓘ
assumes	monotone approximating function in its classical form ⓘ
characterizes	when almost all real numbers admit infinitely many rational approximations of prescribed quality ⓘ
concerns	Lebesgue measure of sets of well-approximable numbers ⓘ approximation of real numbers by rationals ⓘ metric Diophantine approximation ⓘ
criterionType	convergence–divergence criterion ⓘ
domain	real numbers ⓘ
field	Diophantine approximation NERFINISHED ⓘ number theory ⓘ
generalizationOf	Borel–Cantelli lemma applications in Diophantine approximation NERFINISHED ⓘ
givesCriterionFor	existence of infinitely many good rational approximations for almost all real numbers ⓘ
hasConsequence	for almost all real numbers the quality of rational approximation is governed by a simple series test ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	if a certain series converges then the corresponding limsup set has Lebesgue measure zero ⓘ if a certain series diverges then the corresponding limsup set has full Lebesgue measure ⓘ
mathematicalSubjectClassification	11J83 ⓘ 11K60 ⓘ
namedAfter	Aleksandr Yakovlevich Khinchin NERFINISHED ⓘ
quantifier	almost all real numbers ⓘ
relatedTo	Borel–Cantelli lemma NERFINISHED ⓘ Duffin–Schaeffer conjecture NERFINISHED ⓘ Jarník–Besicovitch theorem NERFINISHED ⓘ Khintchine–Groshev theorem NERFINISHED ⓘ
relates	sum of q times approximating function to measure of well-approximable numbers ⓘ
statementForm	zero–one law for Lebesgue measure ⓘ
usedIn	metric theory of Diophantine approximation ⓘ study of well-approximable numbers ⓘ
usesConcept	Lebesgue measure NERFINISHED ⓘ approximating function ⓘ convergence of series ⓘ divergence of series ⓘ limsup set ⓘ

Referenced by (3)

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

Diophantine approximation → hasKeyResult → Khintchine theorem ⓘ

this entity surface form: Khintchine–Groshev theorem

Khinchin's constant → appearsIn → Khintchine theorem ⓘ

this entity surface form: Khinchin's theorem on continued fractions