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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: Joseph L. Gerver
Journal: Proc. Amer. Math. Soc. 62 (1977), 211-214
MSC: Primary 10L10
DOI: https://doi.org/10.1090/S0002-9939-1977-0439796-9
MathSciNet review: 0439796
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0439796-9
Keywords: Arithmetic progression, sum of reciprocals, asymptotic density, Szemerédi's theorem, Rankin's sets
Article copyright: © Copyright 1977 American Mathematical Society