Diophantine approximation

All labels observed (7)

Label	Occurrences
Diophantine approximation canonical	21
metric Diophantine approximation	2
Diophantine approximation language	1
Diophantine approximation theory	1
Diophantine exponent	1
Metric Number Theory	1
Thue–Siegel–Roth theorem	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]

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
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

Referenced by (28)

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

Harold Davenport → areaOfResearch → Diophantine approximation ⓘ

Hardy–Littlewood circle method → requires → Diophantine approximation ⓘ

Diophantine approximation → hasKeyResult → Thue–Siegel–Roth theorem ⓘ

linked to: Diophantine approximation

Diophantine approximation → hasKeyConcept → Diophantine exponent ⓘ

linked to: Diophantine approximation

Diophantine approximation → hasSubfield → metric Diophantine approximation ⓘ

linked to: Diophantine approximation

Johan Frederik Koksma → hasAcademicDiscipline → Diophantine approximation ⓘ

Carl Ludwig Siegel → fieldOfWork → Diophantine approximation ⓘ

Continued Fractions → subject → Diophantine approximation ⓘ

Bombieri–Lang conjecture → expressedIn → Diophantine approximation language ⓘ

linked to: Diophantine approximation

geometry of numbers → appliesTo → Diophantine approximation ⓘ

Hurwitz theorem → field → Diophantine approximation ⓘ

Khintchine theorem → field → Diophantine approximation ⓘ

Jarník–Besicovitch theorem → field → Diophantine approximation ⓘ

Liouville number → relatedTo → Diophantine approximation ⓘ

subject linked to: Liouville numbers

Kurt Mahler → areaOfInfluence → Diophantine approximation ⓘ

An Introduction to Diophantine Approximation → field → Diophantine approximation ⓘ

An Introduction to Diophantine Approximation → topic → metric Diophantine approximation ⓘ

linked to: Diophantine approximation

Farey sequence → relatedTo → Diophantine approximation ⓘ

Stern–Brocot tree → application → Diophantine approximation ⓘ

J. W. S. Cassels → fieldOfWork → Diophantine approximation ⓘ

subject linked to: John

David Masser → researchArea → Diophantine approximation ⓘ

Transcendental Number Theory → focusesOn → Diophantine approximation ⓘ

Glyn Harman → notableWork → Metric Number Theory ⓘ

linked to: Diophantine approximation

Arakelov theory → relatedTo → Diophantine approximation ⓘ

Siegel's theorem on integral points → usesConcept → Diophantine approximation ⓘ

theory of uniform distribution modulo 1 → usesConcept → Diophantine approximation ⓘ

theory of uniform distribution modulo 1 → relatedTo → Diophantine approximation theory ⓘ

linked to: Diophantine approximation

Liouville's inequality in Diophantine approximation → field → Diophantine approximation ⓘ

All labels observed (7)

How this entity was disambiguated Show

Statements (55)

How these facts were elicited Show

Referenced by (28)

How this entity was disambiguated

How these facts were elicited