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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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 ⓘ

Referenced by (1)

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

Diophantine approximation → hasKeyResult → Dirichlet approximation theorem ⓘ