Dirichlet approximation theorem

E637303

mathematical theorem result in Diophantine approximation

The Dirichlet approximation theorem is a fundamental result in Diophantine approximation that guarantees, for any real number and positive integer, the existence of a nearby rational number with bounded denominator and small approximation error.

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All labels observed (3)

Label	Occurrences
Dirichlet approximation theorem canonical	2
Dirichlet's approximation theorem	1
Dirichlet's theorem on Diophantine approximation on manifolds (in basic form)	1

How this entity was disambiguated

This entity first appeared as the object of triple T7030797 — 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: Dirichlet approximation theorem
Context triple: [Diophantine approximation, hasKeyResult, Dirichlet approximation theorem]

A. 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.
B. Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
C. Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
D. Continued Fractions
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.
E. 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.
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: Dirichlet approximation theorem
Target entity description: The Dirichlet approximation theorem is a fundamental result in Diophantine approximation that guarantees, for any real number and positive integer, the existence of a nearby rational number with bounded denominator and small approximation error.

A. 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.
B. Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
C. Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
D. Continued Fractions
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.
E. 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.
F. None of above. chosen

Statements (41)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in Diophantine approximation ⓘ
appearsIn	advanced undergraduate number theory courses ⓘ graduate courses on analytic number theory ⓘ introductory texts on Diophantine approximation ⓘ
category	theorem in analytic number theory ⓘ
concerns	approximation of real numbers by rationals ⓘ bounds on denominators of approximating fractions ⓘ
coreConcept	Diophantine inequality ⓘ fractional parts of multiples of a real number ⓘ lattice points in the plane ⓘ
ensures	existence of p and q with small \|qα − p\| ⓘ
equivalentFormulation	For any real α and positive integer N, there exist integers p and q with 1 ≤ q ≤ N such that \|qα − p\| < 1/N. ⓘ
errorBound	approximation error less than 1/(qN) ⓘ approximation error less than 1/q² for infinitely many rationals ⓘ
field	Diophantine approximation ⓘ number theory ⓘ
generalizationOf	basic rational approximation results from continued fractions ⓘ
guarantees	existence of good rational approximations to real numbers ⓘ
hasVersion	inhomogeneous approximation form ⓘ simultaneous approximation for vectors in R^n ⓘ
historicalPeriod	19th century mathematics ⓘ
holdsFor	every positive integer N ⓘ every real number ⓘ
implies	existence of infinitely many good rational approximations to any irrational number ⓘ for any real α there are infinitely many rationals p/q with \|α − p/q\| < 1/q² ⓘ
influenced	development of modern Diophantine approximation theory ⓘ
namedAfter	Johann Peter Gustav Lejeune Dirichlet ⓘ linked to: Peter Gustav Lejeune Dirichlet
prerequisiteFor	Dirichlet's theorem on Diophantine approximation on manifolds (in basic form) ⓘ linked to: Dirichlet approximation theorem
provides	quantitative bound on rational approximation error ⓘ
relatedTo	Hurwitz's theorem ⓘ linked to: Hurwitz theorem Kronecker approximation theorem ⓘ Minkowski's theorem in geometry of numbers ⓘ linked to: Minkowski’s theorem on convex sets Roth's theorem ⓘ linked to: Roth theorem
statement	For any real number α and any positive integer N, there exist integers p and q with 1 ≤ q ≤ N such that \|α − p/q\| < 1/(qN). ⓘ
strength	gives optimal exponent 2 in \|α − p/q\| < C/q² for general real numbers ⓘ
usedIn	geometry of numbers ⓘ metric Diophantine approximation ⓘ study of continued fractions ⓘ uniform distribution theory ⓘ
usesMethod	pigeonhole principle ⓘ

How these facts were elicited

Referenced by (4)

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

Diophantine approximation → hasKeyResult → Dirichlet approximation theorem ⓘ

Dirichlet approximation theorem → prerequisiteFor → Dirichlet's theorem on Diophantine approximation on manifolds (in basic form) ⓘ

linked to: Dirichlet approximation theorem

Hurwitz theorem → refines → Dirichlet approximation theorem ⓘ

Roth's theorem → comparesWith → Dirichlet's approximation theorem ⓘ

subject linked to: Roth theorem