Erdős discrepancy problem
E554302
The Erdős discrepancy problem is a famous question in combinatorial number theory that asks whether every infinite ±1 sequence has arbitrarily large discrepancy along some homogeneous arithmetic progression.
All labels observed (3)
How this entity was disambiguated
This entity first appeared as the object of triple T5896709 — 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: Erdős discrepancy problem Context triple: [Pál Erdős, knownFor, Erdős discrepancy problem]
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A.
Green–Tao theorem
The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
-
B.
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.
-
C.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
D.
Conway's 99-graph problem
Conway's 99-graph problem is an unsolved combinatorial question in graph theory, posed by John H. Conway, concerning the existence and properties of a hypothetical 99-vertex graph with highly constrained adjacency conditions.
-
E.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
- 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: Erdős discrepancy problem Target entity description: The Erdős discrepancy problem is a famous question in combinatorial number theory that asks whether every infinite ±1 sequence has arbitrarily large discrepancy along some homogeneous arithmetic progression.
-
A.
Green–Tao theorem
The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
-
B.
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.
-
C.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
D.
Conway's 99-graph problem
Conway's 99-graph problem is an unsolved combinatorial question in graph theory, posed by John H. Conway, concerning the existence and properties of a hypothetical 99-vertex graph with highly constrained adjacency conditions.
-
E.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical problem
ⓘ
open problem in mathematics ⓘ |
| asksAbout |
discrepancy of ±1 sequences
ⓘ
homogeneous arithmetic progressions ⓘ |
| concerns | unboundedness of certain partial sums ⓘ |
| coreQuestion | whether every infinite ±1 sequence has unbounded discrepancy on some homogeneous arithmetic progression ⓘ |
| difficulty | hard ⓘ |
| difficultyClassification | very difficult problem in combinatorial number theory ⓘ |
| equivalentFormulation | for every ±1 sequence (x_n) and every C > 0 there exist n,d with |x_d + x_{2d} + … + x_{nd}| > C ⓘ |
| field |
combinatorial number theory
ⓘ
discrepancy theory ⓘ |
| formulation | for every function f: ℕ → {−1, +1} and every C > 0 there exist n,d ∈ ℕ such that |∑_{k=1}^{n} f(kd)| > C ⓘ |
| hasConsequence | every infinite ±1 sequence has arbitrarily large discrepancy along some homogeneous arithmetic progression ⓘ |
| hasOnlinePolymathProject |
Polymath5
ⓘ
linked to:
Polymath Project
|
| hasVariant |
Erdős discrepancy problem for completely multiplicative functions
ⓘ
linked to:
Erdős discrepancy problem
|
| implies | no infinite ±1 sequence has bounded discrepancy on all homogeneous arithmetic progressions ⓘ |
| influencedBy | classical problems of Erdős in additive and combinatorial number theory ⓘ |
| involvesQuantifiers | for all C > 0 there exist n,d ∈ ℕ ⓘ |
| motivation | understanding irregularities of distribution in sequences ⓘ |
| namedAfter |
Paul Erdős
ⓘ
linked to:
Pál Erdős
|
| proposedBy |
Paul Erdős
ⓘ
linked to:
Pál Erdős
|
| publication |
Terence Tao’s 2016 paper in Journal d’Analyse Mathématique
ⓘ
linked to:
Erdős discrepancy problem
|
| quantifiesOver |
all infinite ±1 sequences
ⓘ
all positive integers C ⓘ positive integers d ⓘ positive integers n ⓘ |
| relatedTo |
Erdős–Turán conjecture
ⓘ
completely multiplicative functions ⓘ discrepancy of sequences ⓘ multiplicative functions ⓘ |
| solutionMethod |
Fourier analysis
ⓘ
analytic number theory ⓘ entropy decrement argument ⓘ probabilistic methods ⓘ |
| solvedBy | Terence Tao ⓘ |
| status | solved ⓘ |
| studiedIn |
Polymath5 project
ⓘ
linked to:
Polymath Project
|
| topic |
arithmetic progressions
ⓘ
infinite sequences ⓘ partial sums ⓘ |
| usesConcept |
discrepancy
ⓘ
homogeneous arithmetic progression ⓘ ±1 sequence ⓘ |
| yearProposed | 1930s ⓘ |
| yearSolved | 2015 ⓘ |
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: Erdős discrepancy problem Description of subject: The Erdős discrepancy problem is a famous question in combinatorial number theory that asks whether every infinite ±1 sequence has arbitrarily large discrepancy along some homogeneous arithmetic progression.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
Erdős discrepancy problem
→
hasVariant
→
Erdős discrepancy problem for completely multiplicative functions
ⓘ
linked to: Erdős discrepancy problem
Erdős discrepancy problem
→
publication
→
Terence Tao’s 2016 paper in Journal d’Analyse Mathématique
ⓘ
linked to: Erdős discrepancy problem