Liouville's inequality in Diophantine approximation

E898513

inequality in number theory mathematical theorem result in Diophantine approximation

Liouville's inequality in Diophantine approximation is a foundational result that gives explicit lower bounds on how closely algebraic numbers can be approximated by rationals, leading to the first examples of transcendental numbers.

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Statements (47)

Predicate	Object
instanceOf	inequality in number theory ⓘ mathematical theorem ⓘ result in Diophantine approximation ⓘ
appearsIn	textbooks on Diophantine approximation ⓘ textbooks on transcendental number theory ⓘ
appliesTo	complex algebraic numbers ⓘ real algebraic numbers of degree at least 2 ⓘ
category	Liouville-type inequality ⓘ
classicalReference	Liouville's original papers on transcendental numbers ⓘ
concerns	algebraic numbers ⓘ rational approximations ⓘ
field	Diophantine approximation NERFINISHED ⓘ number theory ⓘ transcendental number theory ⓘ
foundationFor	construction of Liouville numbers ⓘ first explicit examples of transcendental numbers ⓘ
generalFormUses	degree of an algebraic number ⓘ height of an algebraic number ⓘ
gives	lower bounds on approximation of algebraic numbers by rationals ⓘ
hasVariant	inequality for linear forms in algebraic numbers ⓘ inequality for values of polynomials at algebraic points ⓘ
historicalRole	first general result giving explicit irrationality measures for algebraic numbers ⓘ
implies	algebraic numbers have finite irrationality exponent ⓘ algebraic numbers of degree at least 2 cannot be too well approximated by rationals ⓘ existence of transcendental numbers ⓘ no algebraic number of degree at least 2 can be a Liouville number ⓘ
inspired	later metric results in Diophantine approximation ⓘ
introducedBy	Joseph Liouville NERFINISHED ⓘ
introducedInCentury	19th century ⓘ
isPartOf	classical theory of Diophantine approximation ⓘ
isWeakerThan	Roth's theorem NERFINISHED ⓘ Thue–Siegel–Roth theorem NERFINISHED ⓘ
logicalForm	inequality involving absolute values and powers of denominators ⓘ
motivation	understanding how well algebraic numbers can be approximated by rationals ⓘ
namedAfter	Joseph Liouville NERFINISHED ⓘ
provides	explicit constant depending on the algebraic number ⓘ
relatedConcept	Baker's theory of linear forms in logarithms NERFINISHED ⓘ Liouville numbers NERFINISHED ⓘ Roth's theorem NERFINISHED ⓘ Thue–Siegel–Roth theorem NERFINISHED ⓘ irrationality measure ⓘ
statesRoughly	if α is algebraic of degree d ≥ 2 then \|α − p/q\| ≥ C(α)/q^d for all rationals p/q ⓘ
toolIn	elementary transcendence proofs ⓘ proving lower bounds for linear forms in logarithms ⓘ
typeOfBound	lower bound on \|α − p/q\| in terms of q ⓘ
usedFor	proving transcendence of certain real numbers ⓘ
usedToShow	certain rapidly approximable numbers are transcendental ⓘ

Referenced by (1)

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

Joseph Liouville → notableWork → Liouville's inequality in Diophantine approximation ⓘ