Erdős conjecture on arithmetic progressions

Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics (not to be confused with the Erdős–Turán conjecture on additive bases). 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, the conjecture states that if A is a large set in the sense that

$${\displaystyle \sum _{n\in A}{\frac {1}{n}}\ =\ \infty ,}$$

then A contains arithmetic progressions of any given length, meaning that for every positive integer k there are an integer a and a non-zero integer c such that ${\displaystyle \{a,a{+}c,a{+}2c,\ldots ,a{+}kc\}\subset A}$.

In 1936, Erdős and Turán made the weaker conjecture that any set of integers with positive natural density contains infinitely many three-term arithmetic progressions. This was proven by Klaus Roth in 1952, and generalized to arbitrarily long arithmetic progressions by Szemerédi in 1975 in what is now known as Szemerédi's theorem.

In a 1976 talk titled "To the memory of my lifelong friend and collaborator Paul Turán," Paul Erdős offered a prize of US$3000 for a proof of this conjecture. As of 2008 the problem is worth US$5000.

Erdős' conjecture on arithmetic progressions can be viewed as a stronger version of Szemerédi's theorem. Because the sum of the reciprocals of the primes diverges, the Green–Tao theorem on arithmetic progressions is a special case of the conjecture.

The weaker claim that A must contain infinitely many arithmetic progressions of length 3 is a consequence of an improved bound in Roth's theorem. A 2016 paper by Bloom proved that if ${\displaystyle A\subset \{1,..,N\}}$contains no non-trivial three-term arithmetic progressions then ${\displaystyle |A|\ll N(\log {\log {N}})/\log {N}}$.

In 2020 a preprint by Bloom and Sisask improved the bound to ${\displaystyle |A|\ll N/(\log {N})^{1+c}}$for some absolute constant ${\displaystyle c>0}$.

In 2023 a new bound of ${\displaystyle \exp(-c(\log N)^{1/12})N}$was found by computer scientists Kelley and Meka, with an exposition given in more familiar mathematical language by Bloom and Sisask, who have since improved the exponent of the Kelly-Meka bound to ${\displaystyle \beta =1/9}$, and conjectured ${\displaystyle \beta =5/41}$, in a preprint.

• Problems involving arithmetic progressions
• List of sums of reciprocals
• List of conjectures by Paul Erdős
• Müntz–Szász theorem

• 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 and P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.
• 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.doi:10.1007/BF02579174

• The Erdős–Turán conjecture or the Erdős conjecture? on MathOverflow