theory of uniform distribution modulo 1
E824093
The theory of uniform distribution modulo 1 is a branch of number theory that studies how sequences of real numbers distribute their fractional parts evenly in the unit interval.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Borel’s normal number theorem | 1 |
| theory of uniform distribution modulo 1 canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9838938 — 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: theory of uniform distribution modulo 1 Context triple: [Johan Frederik Koksma, contributedTo, theory of uniform distribution modulo 1]
-
A.
Khintchine theorem
Khintchine theorem is a fundamental result in metric Diophantine approximation that characterizes, via a simple convergence–divergence criterion, when almost all real numbers admit infinitely many rational approximations of a prescribed quality.
-
B.
Erdős–Wintner theorem
The Erdős–Wintner theorem is a fundamental result in probabilistic number theory that characterizes when an additive arithmetic function has a limiting distribution.
-
C.
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.
-
D.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
E.
Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
- 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: theory of uniform distribution modulo 1 Target entity description: The theory of uniform distribution modulo 1 is a branch of number theory that studies how sequences of real numbers distribute their fractional parts evenly in the unit interval.
-
A.
Khintchine theorem
Khintchine theorem is a fundamental result in metric Diophantine approximation that characterizes, via a simple convergence–divergence criterion, when almost all real numbers admit infinitely many rational approximations of a prescribed quality.
-
B.
Erdős–Wintner theorem
The Erdős–Wintner theorem is a fundamental result in probabilistic number theory that characterizes when an additive arithmetic function has a limiting distribution.
-
C.
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.
-
D.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
E.
Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
- F. None of above. chosen
Statements (55)
| Predicate | Object |
|---|---|
| instanceOf |
branch of number theory
ⓘ
mathematical theory ⓘ |
| appliesTo |
Halton sequences
NERFINISHED
ⓘ
Sobol’ sequences NERFINISHED ⓘ multidimensional sequences in [0,1)^s ⓘ polynomial sequences (P(n)) modulo 1 ⓘ sequences (nα) modulo 1 ⓘ sequences (x_n) of real numbers ⓘ van der Corput sequences NERFINISHED ⓘ |
| basedOn |
Birkhoff ergodic theorem
NERFINISHED
ⓘ
Erdős–Turán inequality NERFINISHED ⓘ Koksma–Hlawka inequality NERFINISHED ⓘ Kronecker’s theorem NERFINISHED ⓘ Weyl criterion NERFINISHED ⓘ |
| developedBy |
Harald Bohr
NERFINISHED
ⓘ
Hermann Weyl NERFINISHED ⓘ Johannes van der Corput NERFINISHED ⓘ Mark Kac NERFINISHED ⓘ Paul Erdős NERFINISHED ⓘ |
| fieldOfStudy | uniform distribution modulo 1 ⓘ |
| hasApplicationIn |
discrepancy theory
NERFINISHED
ⓘ
numerical integration ⓘ quasi-Monte Carlo integration ⓘ randomized algorithms ⓘ |
| hasKeyResult |
Erdős–Turán inequality for discrepancy
NERFINISHED
ⓘ
Koksma–Hlawka inequality NERFINISHED ⓘ Kronecker’s theorem on inhomogeneous approximation NERFINISHED ⓘ Weyl criterion for uniform distribution NERFINISHED ⓘ Weyl’s theorem on polynomial sequences NERFINISHED ⓘ metric theorems on normal numbers ⓘ results on lacunary sequences ⓘ van der Corput difference theorem NERFINISHED ⓘ |
| relatedTo |
Diophantine approximation theory
NERFINISHED
ⓘ
ergodic theory NERFINISHED ⓘ normal numbers ⓘ probability theory NERFINISHED ⓘ pseudorandom number generation ⓘ quasi-Monte Carlo methods NERFINISHED ⓘ |
| studies |
Diophantine approximation aspects of sequences
ⓘ
Kronecker sequences NERFINISHED ⓘ Weyl sums NERFINISHED ⓘ discrepancy of sequences ⓘ distribution of fractional parts of sequences ⓘ equidistribution of sequences in the unit interval ⓘ exponential sums ⓘ low-discrepancy sequences ⓘ metric distribution properties of sequences ⓘ |
| usesConcept |
Diophantine approximation
NERFINISHED
ⓘ
Fourier analysis NERFINISHED ⓘ discrepancy function ⓘ equidistribution ⓘ exponential functions e^{2πinx} ⓘ fractional part of a real number ⓘ measure theory ⓘ unit interval [0,1) ⓘ |
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: theory of uniform distribution modulo 1 Description of subject: The theory of uniform distribution modulo 1 is a branch of number theory that studies how sequences of real numbers distribute their fractional parts evenly in the unit interval.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Borel’s normal number theorem