Erdős–Turán conjecture

E554303

mathematical conjecture problem in additive number theory unsolved problem in mathematics

The Erdős–Turán conjecture is an unsolved problem in additive number theory asserting that any subset of the positive integers with divergent sum of reciprocals must contain arbitrarily long arithmetic progressions.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Erdős–Turán conjecture on arithmetic progressions	1

Statements (47)

Predicate	Object
instanceOf	mathematical conjecture ⓘ problem in additive number theory ⓘ unsolved problem in mathematics ⓘ
coAuthor	Paul Erdős NERFINISHED ⓘ Pál Turán NERFINISHED ⓘ
concerns	arithmetic progressions ⓘ divergent series of reciprocals ⓘ subsets of the positive integers ⓘ
conclusion	the subset contains arithmetic progressions of every finite length ⓘ
condition	the sum of reciprocals of the elements of the subset diverges ⓘ
contrastWith	Szemerédi's theorem, which uses combinatorial density conditions NERFINISHED ⓘ
difficulty	considered very difficult ⓘ
doesNotRequire	positive asymptotic density of the subset ⓘ
domainRestriction	subsets of the positive integers ⓘ
field	additive number theory ⓘ number theory ⓘ
hasAbbreviation	Erdős–Turán conjecture on arithmetic progressions NERFINISHED ⓘ
hasConsequence	would generalize many known results on arithmetic progressions in special sets if proved ⓘ
hasFormulation	If A is a subset of the positive integers and sum_{a in A} 1/a diverges, then A contains arithmetic progressions of every finite length. ⓘ
implies	existence of 3-term arithmetic progressions under the divergence condition ⓘ existence of arbitrarily long arithmetic progressions ⓘ existence of k-term arithmetic progressions for every positive integer k under the divergence condition ⓘ
involves	infinite subsets of the positive integers ⓘ
language	stated in the language of additive combinatorics ⓘ
motivatedBy	study of additive structure in large sets of integers ⓘ
namedAfter	Paul Erdős NERFINISHED ⓘ Pál Turán NERFINISHED ⓘ
namedEntity	Erdős–Turán conjecture NERFINISHED ⓘ
openAsOf	2024 ⓘ
quantifier	arbitrarily long arithmetic progressions ⓘ
relatedTo	Erdős discrepancy problem NERFINISHED ⓘ Erdős–Turán inequality NERFINISHED ⓘ Green–Tao theorem NERFINISHED ⓘ Szemerédi's theorem NERFINISHED ⓘ problems on sets of multiples and additive bases ⓘ
requires	divergence of the harmonic sum over the subset ⓘ
statementInformal	Any subset of the positive integers whose sum of reciprocals diverges contains arbitrarily long arithmetic progressions. ⓘ
status	open ⓘ
strongerThan	Green–Tao theorem for primes NERFINISHED ⓘ assertions about existence of only finitely long progressions ⓘ
topic	arithmetic progressions in dense sets ⓘ density conditions for arithmetic progressions ⓘ
typeOfCondition	analytic density condition ⓘ
usesConcept	arithmetic progression ⓘ divergent series ⓘ reciprocal sums ⓘ
yearProposed	1936 ⓘ

Referenced by (2)

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

Pál Erdős → knownFor → Erdős–Turán conjecture ⓘ

Green–Tao theorem → relatedTo → Erdős–Turán conjecture ⓘ

this entity surface form: Erdős–Turán conjecture on arithmetic progressions