van der Corput inequality
E776135
The van der Corput inequality is a fundamental result in analytic number theory that provides bounds for exponential sums, playing a key role in estimating trigonometric sums and studying uniform distribution.
All labels observed (1)
| Label | Occurrences |
|---|---|
| van der Corput inequality canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T9070946 — 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: van der Corput inequality Context triple: [Johannes G. van der Corput, knownFor, van der Corput inequality]
-
A.
Korn inequality
Korn inequality is a fundamental result in functional analysis and the mathematical theory of elasticity that provides bounds relating the full gradient of a vector field to its symmetric part, ensuring control of deformations by their strains.
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B.
Bernstein inequalities
Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.
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C.
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality is a fundamental result in linear algebra and analysis that bounds the inner product of two vectors by the product of their magnitudes, underpinning many concepts in geometry, probability, and functional analysis.
-
D.
Turán–Kubilius inequality
The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
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E.
Hadamard inequality
The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
- 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: van der Corput inequality Target entity description: The van der Corput inequality is a fundamental result in analytic number theory that provides bounds for exponential sums, playing a key role in estimating trigonometric sums and studying uniform distribution.
-
A.
Korn inequality
Korn inequality is a fundamental result in functional analysis and the mathematical theory of elasticity that provides bounds relating the full gradient of a vector field to its symmetric part, ensuring control of deformations by their strains.
-
B.
Bernstein inequalities
Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.
-
C.
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality is a fundamental result in linear algebra and analysis that bounds the inner product of two vectors by the product of their magnitudes, underpinning many concepts in geometry, probability, and functional analysis.
-
D.
Turán–Kubilius inequality
The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
-
E.
Hadamard inequality
The Hadamard inequality is a fundamental result in linear algebra and analysis that bounds the absolute value of a determinant by the product of the Euclidean norms of its row or column vectors.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
tool in the theory of exponential sums ⓘ |
| appliesTo |
finite exponential sums
ⓘ
sequences of complex numbers ⓘ trigonometric polynomials ⓘ |
| field |
analytic number theory
ⓘ
harmonic analysis ⓘ uniform distribution theory ⓘ |
| hasConsequence |
bounds for discrepancy of sequences in [0,1)
ⓘ
error term estimates in equidistribution theorems ⓘ improved bounds for exponential sums with smooth phase functions ⓘ |
| hasForm | relates square of absolute value of an exponential sum to sums of correlations of coefficients ⓘ |
| hasVariant |
first derivative test
ⓘ
higher derivative tests ⓘ second derivative test ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| involvesConcept |
correlation sums
ⓘ
discrepancy theory ⓘ equidistribution ⓘ exponential sums ⓘ trigonometric sums ⓘ uniform distribution modulo 1 ⓘ |
| isPartOf | van der Corput method for exponential sums ⓘ |
| namedAfter |
J. G. van der Corput
ⓘ
linked to:
Johannes G. van der Corput
|
| relatedTo |
Bessel’s inequality
ⓘ
linked to:
Bessel inequality
Cauchy–Schwarz inequality ⓘ Weyl differencing ⓘ Weyl’s inequality ⓘ
linked to:
Weyl inequalities
large sieve inequality ⓘ van der Corput method ⓘ |
| typicalStatement | gives an upper bound for |∑_{n=1}^N a_n e(f(n))|^2 in terms of sums over shifts h of ∑_{n} a_{n+h} \overline{a_n} ⓘ |
| usedFor |
bounding Weyl sums
ⓘ
bounding exponential sums ⓘ bounding exponential sums over integers ⓘ bounding partial sums of multiplicative functions ⓘ estimating discrepancy of sequences ⓘ estimating trigonometric sums ⓘ proving equidistribution results ⓘ studying uniform distribution modulo 1 ⓘ |
| usedIn |
bounds for character sums
ⓘ
bounds for exponential sums in prime number theory ⓘ distribution of sequences like (nα) mod 1 ⓘ estimates for exponential sums with polynomial phases ⓘ estimates in the circle method ⓘ metric number theory ⓘ proofs of Weyl’s criterion applications ⓘ |
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: van der Corput inequality Description of subject: The van der Corput inequality is a fundamental result in analytic number theory that provides bounds for exponential sums, playing a key role in estimating trigonometric sums and studying uniform distribution.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.