Continued Fractions
E379001
Continued Fractions is a classic mathematical monograph by Aleksandr Khinchin that systematically develops the theory and applications of continued fraction expansions in number theory and analysis.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Continued Fractions canonical | 1 |
| Khinchin's theorem on continued fractions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3677830 — 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: Continued Fractions Context triple: [Aleksandr Khinchin, notableWork, Continued Fractions]
-
A.
Farey tessellation
The Farey tessellation is a geometric partition of the hyperbolic plane into ideal triangles whose vertices correspond to rational numbers, closely linked to number theory and modular group actions.
-
B.
Diophantine approximation
Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
-
C.
Lucas sequences
Lucas sequences are a family of integer sequences defined by the same type of second-order linear recurrence as the Fibonacci numbers but with more general initial conditions, encompassing the Fibonacci sequence as a special case.
-
D.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
E.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
- 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: Continued Fractions Target entity description: Continued Fractions is a classic mathematical monograph by Aleksandr Khinchin that systematically develops the theory and applications of continued fraction expansions in number theory and analysis.
-
A.
Farey tessellation
The Farey tessellation is a geometric partition of the hyperbolic plane into ideal triangles whose vertices correspond to rational numbers, closely linked to number theory and modular group actions.
-
B.
Diophantine approximation
Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
-
C.
Lucas sequences
Lucas sequences are a family of integer sequences defined by the same type of second-order linear recurrence as the Fibonacci numbers but with more general initial conditions, encompassing the Fibonacci sequence as a special case.
-
D.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
E.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
- F. None of above. chosen
Statements (34)
| Predicate | Object |
|---|---|
| instanceOf |
book
ⓘ
mathematical monograph ⓘ |
| author |
Khinchin
ⓘ
surface form:
A. Ya. Khinchin
Aleksandr Khinchin ⓘ
surface form:
Aleksandr Yakovlevich Khinchin
|
| considered |
classic text on continued fractions
ⓘ
standard reference in metric theory of continued fractions ⓘ |
| emphasizes |
measure-theoretic methods in number theory
ⓘ
probabilistic methods in analysis of continued fractions ⓘ |
| focusesOn |
continued fraction expansions of real numbers
ⓘ
ergodic properties of the Gauss map ⓘ metric properties of continued fractions ⓘ probabilistic aspects of continued fractions ⓘ |
| hasInfluenceOn |
ergodic theory of number-theoretic transformations
ⓘ
modern metric number theory ⓘ probabilistic number theory ⓘ |
| hasMainTopic |
representation of real numbers by continued fractions
ⓘ
statistical properties of continued fraction digits ⓘ |
| includes |
Khinchin's constant
ⓘ
surface form:
Khintchine’s constant
distribution of partial quotients ⓘ metric theorems on continued fractions ⓘ results on approximation of irrationals by rationals ⓘ theory of simple continued fractions ⓘ |
| originalLanguage | Russian ⓘ |
| relatedTo |
Borel normal numbers
ⓘ
Diophantine approximation exponents ⓘ Gauss–Kuzmin distribution ⓘ ergodic theory ⓘ |
| subject |
Diophantine approximation
ⓘ
analysis ⓘ continued fractions ⓘ metric number theory ⓘ number theory ⓘ |
| usedIn |
advanced courses on Diophantine approximation
ⓘ
graduate-level courses in number theory ⓘ |
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Continued Fractions Description of subject: Continued Fractions is a classic mathematical monograph by Aleksandr Khinchin that systematically develops the theory and applications of continued fraction expansions in number theory and analysis.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Khinchin's theorem on continued fractions