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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
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