Erdős-Turán conjecture

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The Erdős-Turán conjecture is an unproven proposition in additive number theory. The conjecture 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

then A contains arithmetic progressions of any given length.

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

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