## Arithmetic progressions in lacunary sets.(English)Zbl 0632.10052

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.
Reviewer: Št.Porubský

### MSC:

 11B25 Arithmetic progressions 11N13 Primes in congruence classes
Full Text: