Jarník–Besicovitch theorem
E637306
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jarník–Besicovitch theorem canonical | 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]
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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.
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
rational approximation of real numbers ⓘ sets of real numbers ⓘ very well approximable real numbers ⓘ |
| field |
Diophantine approximation
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Vojtěch Jarník NERFINISHED ⓘ |
| refines | Khintchine-type results in Diophantine approximation ⓘ |
| relatedTo |
Hausdorff dimension theory
ⓘ
Jarník theorem NERFINISHED ⓘ Khintchine theorem NERFINISHED ⓘ sets of badly approximable numbers ⓘ |
| topic |
distribution of well-approximable numbers
ⓘ
metric properties of Diophantine approximation ⓘ |
| usesConcept |
Hausdorff measure
NERFINISHED
ⓘ
approximation exponent ⓘ limsup sets ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Jarník–Besicovitch theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.