The sum of the reciprocals of a set of integers with no arithmetic progression of $k$ terms

Abstract:

It is shown that for each integer $k \geqslant 3$, there exists a set ${S_k}$ of positive integers containing no arithmetic progression of k terms, such that ${\Sigma _{n \in {S_k}}}1/n > (1 - \varepsilon )k\log k$, with a finite number of exceptional k for each real $\varepsilon > 0$. This result is shown to be superior to that attainable with other sets in the literature, in particular Rankin’s sets $\mathcal {A}(k)$, which have the highest known asymptotic density for sets of positive integers containing no arithmetic progression of k terms.