Arithmetic progressions in lacunary sets (Q1095188)

One of the well-known Erdős conjectures says: If for a set A of positive integers the series \(\sum_{a\in A}a^{-1}\) diverges then A contains a k-term arithmetic progression for all \(k\geq 1\). In the paper some observations regarding this conjecture are made and several special cases of the conjecture are proved. For instance, the following statement is equivalent to the above conjecture: For each positive integer k there exists T such that if \(\sum_{a\in A}a^{-1}>T\) then A contains a k-term arithmetic progression. Or, if A is M-lacunary (i.e. if the sequence of differences of consecutive terms of A is non-decreasing and tends to infinity) and \(\sum_{a\in A}a^{-1}=\infty,\) then A satisfies the Erdős conjecture, etc.