Erdős conjecture on arithmetic progressions

Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture due to Turán's earlier work with Erdős,[1] 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, 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.

In 1936, Erdős and Turán made the weaker conjecture that any set of integers with positive natural density contains arbitrarily long arithmetic progressions.[2] This was proven by Szemerédi in 1975, and is now known as Szemerédi's theorem. Erdős' conjecture on arithmetic progressions can be viewed as a stronger version of Szemerédi's theorem, and if this conjecture were proven, it would imply the Green–Tao theorem on arithmetic progressions in the primes since the sum of the reciprocals of the primes diverges.

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.[3] The problem is currently worth US$5000.[4]

Even the weaker claim, that A must contain at least one arithmetic progression of length 3, is open, and the best known bound is due to Tom Sanders.[5]

The converse of this result is not true, as seen by the set $A = \{1,10,11,100,101,102,1000,1001,1002,1003,...\}$, the sum of the reciprocals of which converges.