Jarník–Besicovitch theorem

E637306

mathematical theorem theorem in metric number theory

The Jarník–Besicovitch theorem is a fundamental result in metric number theory that determines the Hausdorff dimension of sets of real numbers that are very well approximable by rationals.

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All labels observed (2)

Label	Occurrences
Jarník–Besicovitch theorem canonical	2
Jarník theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T7030800 — 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: Jarník–Besicovitch theorem
Context triple: [Diophantine approximation, hasKeyResult, Jarník–Besicovitch theorem]

A. Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
B. Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
C. Helly’s theorem
Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
D. Erdős–Szekeres theorem
The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
E. Radon’s theorem
Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
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: Jarník–Besicovitch theorem
Target entity description: The Jarník–Besicovitch theorem is a fundamental result in metric number theory that determines the Hausdorff dimension of sets of real numbers that are very well approximable by rationals.

A. Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
B. Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
C. Helly’s theorem
Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
D. Erdős–Szekeres theorem
The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
E. Radon’s theorem
Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
F. None of above. chosen

Statements (35)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in metric number theory ⓘ
appliesTo	real numbers ⓘ subsets of the real line ⓘ
area	fractal geometry ⓘ number theory ⓘ
characterizes	Hausdorff dimension of very well approximable numbers ⓘ size of exceptional sets in Diophantine approximation ⓘ
concerns	Hausdorff dimension ⓘ rational approximation of real numbers ⓘ sets of real numbers ⓘ very well approximable real numbers ⓘ
field	Diophantine approximation ⓘ metric number theory ⓘ
hasConsequence	precise dimension formula for very well approximable sets ⓘ
hasImportance	fundamental result in metric number theory ⓘ
implies	sets of very well approximable numbers have Lebesgue measure zero ⓘ
isAbout	quantitative size of sets defined by Diophantine inequalities ⓘ
isUsedIn	analysis of approximation properties of typical real numbers ⓘ study of fractal sets in number theory ⓘ
mathematicalDomain	analysis ⓘ geometric measure theory ⓘ measure theory ⓘ
namedAfter	Abram Samoilovitch Besicovitch ⓘ linked to: A. S. Besicovitch Vojtěch Jarník ⓘ
refines	Khintchine-type results in Diophantine approximation ⓘ
relatedTo	Hausdorff dimension theory ⓘ Jarník theorem ⓘ linked to: Jarník–Besicovitch theorem Khintchine theorem ⓘ sets of badly approximable numbers ⓘ
topic	distribution of well-approximable numbers ⓘ metric properties of Diophantine approximation ⓘ
usesConcept	Hausdorff measure ⓘ approximation exponent ⓘ limsup sets ⓘ

How these facts were elicited

Referenced by (3)

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

Diophantine approximation → hasKeyResult → Jarník–Besicovitch theorem ⓘ

Khintchine theorem → relatedTo → Jarník–Besicovitch theorem ⓘ

Jarník–Besicovitch theorem → relatedTo → Jarník theorem ⓘ

linked to: Jarník–Besicovitch theorem

All labels observed (2)

How this entity was disambiguated Show

Statements (35)

How these facts were elicited Show

Referenced by (3)

How this entity was disambiguated

How these facts were elicited