Erdős conjecture on arithmetic progressions

Erdős' conjecture on arithmetic progressions, often incorrectly referred to as the Erdős–Turán conjecture (see also Erdős–Turán conjecture on additive bases), is a conjecture in additive combinatorics due to Paul Erdős. It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.

Formally, if

$\sum_{n\in A} \frac{1}{n} = \infty$

(i.e. A is a large set) then A contains arithmetic progressions of any given length.

If true, the theorem would generalize Szemerédi's theorem.

Erdős offered a prize of US$3000 for a proof of this conjecture at the time. The problem is currently worth US$5000.

The Green–Tao theorem on arithmetic progressions in the primes is a special case of this conjecture.

References

• P. Erdős: Résultats et problèmes en théorie de nombres, Séminaire Delange-Pisot-Poitou (14e année: 1972/1973), Théorie des nombres, Fasc 2., Exp. No. 24, pp. 7,
• P. Erdős: Problems in number theory and combinatorics, Proc. Sixth Manitoba Conf. on Num. Math., Congress Numer. XVIII(1977), 35–58.
• P. Erdős: On the combinatorial problems which I would most like to see solved, Combinatorica, 1(1981), 28.