Buck’s measure density and sets of positive integers containing arithmetic progression. (English) Zbl 0761.11004

{For notations and related results see the preceding review.} Let \(D_ 2\) be the class of all \(S\subseteq\mathbb{N}\) possessing a natural density \(d(S)\), and let \(D_ 3\) be the class of all “\(d\)-measurable” sets \(S\subseteq\mathbb{N}\). The authors re-prove the result by Buck [loc. cit.] that \(D_ 3=D_ 2\). Furthermore, they give another proof of assertion (i) above. Finally, they show the existence of a set \(S_ 0\), with \(\mu^*(S_ 0)=1\), not containing any three-term arithmetic progression.


11B05 Density, gaps, topology
11B83 Special sequences and polynomials
28E99 Miscellaneous topics in measure theory
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