Diophantine approximation
E163264
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Diophantine approximation canonical | 5 |
| Diophantine exponent | 1 |
| Thue–Siegel–Roth theorem | 1 |
| metric Diophantine approximation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1428733 — 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: Diophantine approximation Context triple: [Harold Davenport, areaOfResearch, Diophantine approximation]
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A.
Lindemann–Weierstrass theorem precursor
The Lindemann–Weierstrass theorem precursor is an early foundational result in transcendental number theory developed by Ferdinand von Lindemann that paved the way for the full Lindemann–Weierstrass theorem on the algebraic independence of exponentials of algebraic numbers.
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B.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
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C.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
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D.
An Introduction to the Theory of Numbers
An Introduction to the Theory of Numbers is a classic textbook in number theory, co-authored by G. H. Hardy, that systematically develops fundamental concepts such as divisibility, prime numbers, Diophantine equations, and quadratic forms.
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E.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
- 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: Diophantine approximation Target entity description: 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.
-
A.
Lindemann–Weierstrass theorem precursor
The Lindemann–Weierstrass theorem precursor is an early foundational result in transcendental number theory developed by Ferdinand von Lindemann that paved the way for the full Lindemann–Weierstrass theorem on the algebraic independence of exponentials of algebraic numbers.
-
B.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
C.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
D.
An Introduction to the Theory of Numbers
An Introduction to the Theory of Numbers is a classic textbook in number theory, co-authored by G. H. Hardy, that systematically develops fundamental concepts such as divisibility, prime numbers, Diophantine equations, and quadratic forms.
-
E.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
- F. None of above. chosen
Statements (55)
| Predicate | Object |
|---|---|
| instanceOf |
branch of number theory
ⓘ
mathematical discipline ⓘ |
| appliesTo |
algebraic numbers
ⓘ
matrices ⓘ real numbers ⓘ vectors in Euclidean space ⓘ |
| fieldOfStudy |
Diophantine equations
ⓘ
Diophantine inequalities ⓘ approximation of real numbers by rational numbers ⓘ metric properties of rational approximations ⓘ |
| goal |
obtain bounds on approximation errors
ⓘ
quantify how well real numbers can be approximated by rationals ⓘ |
| hasKeyConcept |
Diophantine exponent
ⓘ
linked to:
Diophantine approximation
Hausdorff dimension of exceptional sets ⓘ Liouville numbers ⓘ badly approximable numbers ⓘ continued fraction expansion ⓘ irrationality measure ⓘ lattice point counting ⓘ very well approximable numbers ⓘ |
| hasKeyResult |
Baker theorem on linear forms in logarithms
ⓘ
Dirichlet approximation theorem ⓘ Hurwitz theorem ⓘ Jarník–Besicovitch theorem ⓘ Khintchine theorem ⓘ Khintchine–Groshev theorem ⓘ
linked to:
Khintchine theorem
Minkowski convex body theorem ⓘ
linked to:
Minkowski’s theorem on convex sets
Roth theorem ⓘ Subspace theorem ⓘ Thue–Siegel–Roth theorem ⓘ
linked to:
Diophantine approximation
|
| hasSubfield |
inhomogeneous Diophantine approximation
ⓘ
metric Diophantine approximation ⓘ
linked to:
Diophantine approximation
multiplicative Diophantine approximation ⓘ p-adic Diophantine approximation ⓘ uniform Diophantine approximation ⓘ |
| namedAfter | Diophantus of Alexandria ⓘ |
| relatedTo |
Diophantine geometry
ⓘ
continued fractions ⓘ ergodic theory ⓘ geometry of numbers ⓘ homogeneous dynamics ⓘ measure theory ⓘ probability theory ⓘ transcendental number theory ⓘ |
| studies |
approximation by algebraic numbers
ⓘ
approximation by integers ⓘ approximation exponents of real numbers ⓘ inhomogeneous Diophantine approximation ⓘ quality of approximation of real numbers by rationals ⓘ rates of approximation of real numbers by rationals ⓘ simultaneous approximation of several real numbers ⓘ uniform approximation by rationals ⓘ |
| usedIn |
proofs of irrationality results
ⓘ
proofs of transcendence results ⓘ results on distribution modulo 1 ⓘ |
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: Diophantine approximation Description of subject: 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.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Diophantine approximation
linked to: Diophantine approximation
linked to: Diophantine approximation