Liouville's inequality in Diophantine approximation
E898513
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Liouville's inequality in Diophantine approximation canonical | 1 |
| Liouville's theorem on Diophantine approximation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10992153 — 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: Liouville's inequality in Diophantine approximation Context triple: [Joseph Liouville, notableWork, Liouville's inequality in Diophantine approximation]
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A.
An Introduction to Diophantine Approximation
"An Introduction to Diophantine Approximation" is a classic mathematical monograph that systematically develops the theory of approximating real numbers by rationals, aimed at advanced undergraduates and researchers in number theory.
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B.
Baker theorem on linear forms in logarithms
The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.
-
C.
Liouville numbers
Liouville numbers are real numbers that can be approximated extremely closely by rationals, making them a classic example of transcendental numbers in number theory.
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D.
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.
-
E.
Transcendental Number Theory
Transcendental Number Theory is a mathematical monograph by Alan Baker that develops methods for studying transcendental and algebraic numbers, particularly through linear forms in logarithms.
- 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: Liouville's inequality in Diophantine approximation Target entity description: 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.
-
A.
An Introduction to Diophantine Approximation
"An Introduction to Diophantine Approximation" is a classic mathematical monograph that systematically develops the theory of approximating real numbers by rationals, aimed at advanced undergraduates and researchers in number theory.
-
B.
Baker theorem on linear forms in logarithms
The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.
-
C.
Liouville numbers
Liouville numbers are real numbers that can be approximated extremely closely by rationals, making them a classic example of transcendental numbers in number theory.
-
D.
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.
-
E.
Transcendental Number Theory
Transcendental Number Theory is a mathematical monograph by Alan Baker that develops methods for studying transcendental and algebraic numbers, particularly through linear forms in logarithms.
- F. None of above. chosen
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
ⓘ
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 ⓘ |
| introducedInCentury | 19th century ⓘ |
| isPartOf | classical theory of Diophantine approximation ⓘ |
| isWeakerThan |
Roth's theorem
ⓘ
linked to:
Roth theorem
Thue–Siegel–Roth theorem ⓘ |
| logicalForm | inequality involving absolute values and powers of denominators ⓘ |
| motivation | understanding how well algebraic numbers can be approximated by rationals ⓘ |
| namedAfter | Joseph Liouville ⓘ |
| provides | explicit constant depending on the algebraic number ⓘ |
| relatedConcept |
Baker's theory of linear forms in logarithms
ⓘ
linked to:
Baker theorem on linear forms in logarithms
Liouville numbers ⓘ Roth's theorem ⓘ
linked to:
Roth theorem
Thue–Siegel–Roth theorem ⓘ 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 ⓘ |
How these facts were elicited
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Input
Subject: Liouville's inequality in Diophantine approximation Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Roth theorem
linked to: Liouville's inequality in Diophantine approximation