Khintchine theorem
E637305
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.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Khintchine theorem canonical | 2 |
| Duffin–Schaeffer conjecture | 1 |
| Khinchin theorem | 1 |
| Khinchin's theorem on continued fractions | 1 |
| Khinchin-type theorems in Diophantine approximation | 1 |
| Khintchine–Groshev theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7030799 — 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: Khintchine theorem Context triple: [Diophantine approximation, hasKeyResult, Khintchine theorem]
-
A.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
-
B.
Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
-
C.
Khinchin–Lévy constant
The Khinchin–Lévy constant is a mathematical constant arising in metric number theory and continued fractions, describing the typical exponential growth rate of the denominators of convergents for almost all real numbers.
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D.
Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
-
E.
Khinchin's law of the iterated logarithm
Khinchin's law of the iterated logarithm is a fundamental result in probability theory that precisely characterizes the almost-sure fluctuations of partial sums of independent random variables on the scale of the square root of twice the product of their variance and the iterated logarithm of the sample size.
- 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: Khintchine theorem Target entity description: 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.
-
A.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
-
B.
Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
-
C.
Khinchin–Lévy constant
The Khinchin–Lévy constant is a mathematical constant arising in metric number theory and continued fractions, describing the typical exponential growth rate of the denominators of convergents for almost all real numbers.
-
D.
Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
-
E.
Khinchin's law of the iterated logarithm
Khinchin's law of the iterated logarithm is a fundamental result in probability theory that precisely characterizes the almost-sure fluctuations of partial sums of independent random variables on the scale of the square root of twice the product of their variance and the iterated logarithm of the sample size.
- F. None of above. chosen
Statements (38)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in metric Diophantine approximation ⓘ |
| alternativeName |
Khinchin theorem
ⓘ
linked to:
Khintchine theorem
|
| 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
ⓘ
number theory ⓘ |
| generalizationOf | Borel–Cantelli lemma applications in Diophantine approximation ⓘ |
| 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
ⓘ
linked to:
Aleksandr Khinchin
|
| quantifier | almost all real numbers ⓘ |
| relatedTo |
Borel–Cantelli lemma
ⓘ
linked to:
Borel–Cantelli lemmas
Duffin–Schaeffer conjecture ⓘ
linked to:
Khintchine theorem
Jarník–Besicovitch theorem ⓘ Khintchine–Groshev theorem ⓘ |
| 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
ⓘ
approximating function ⓘ convergence of series ⓘ divergence of series ⓘ limsup set ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Khintchine theorem Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Khintchine theorem
linked to: Khintchine theorem
linked to: Khintchine theorem
linked to: Khintchine theorem
linked to: Khintchine theorem