Dirichlet approximation theorem
E637303
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Dirichlet approximation theorem canonical | 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.
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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.
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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.
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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.
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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 NERFINISHED ⓘ |
| prerequisiteFor | Dirichlet's theorem on Diophantine approximation on manifolds (in basic form) NERFINISHED ⓘ |
| provides | quantitative bound on rational approximation error ⓘ |
| relatedTo |
Hurwitz's theorem
NERFINISHED
ⓘ
Kronecker approximation theorem NERFINISHED ⓘ Minkowski's theorem in geometry of numbers NERFINISHED ⓘ Roth's theorem NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
metric Diophantine approximation ⓘ study of continued fractions ⓘ uniform distribution theory ⓘ |
| usesMethod | pigeonhole principle NERFINISHED ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Dirichlet approximation theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.