Nonstandard methods and the Erdős-Turán conjecture

Abstract

A major open question in combinatorial number theory is the Erdős-Turán conjecture which states that if A = 〈a _n〉 is a sequence of natural numbers with the property that ∑ ^∞ _n=1 1/a _n diverges then A contains arbitrarily long arithmetic progressions [1]. The difficulty of this problem is underscored by the fact that a positive answer would generalize Szcmerédi’s theorem which says that if a sequence A⊂ ℕ has positive upper Banach Density then A contains arbitrarily long arithmetic progressions. Szemerédi’s theorem itself has been the object of intense interest, since first, conjectured, also by Erdős and Turán, in 1936. First proved by Szemerédi in 1974 [9], the theorem has been re-proved using completely different approaches by Furstenberg in 1977 [2]