Ideals of nowhere dense sets in some topologies on positive integers
The authors study three topologies on \(\mathbb{N}\), the set of natural numbers, defined by taking families of arithmetic progressions as a base. These were defined by \textit{H. Furstenberg} [Am. Math. Mon. 62, No. 5, 353 (1955; Zbl 1229.11009)], \textit{S. W. Golomb} [Am. Math. Mon. 66, 663--665 (1959; Zbl 0202.33001)], and \textit{A. M. Kirch} [Am. Math. Mon. 76, 169--171 (1969; Zbl 0174.25602)]. Furstenberg used all arithmetic progressions \(\{b+an:n\in\mathbb{N}_0\}\) with \(b\leq a\), Golomb demanded \(b<a\) and \(\text{gcd}(a,b)=1\), and Kirch added the demand that \(a\) be square-free to Golomb's conditions. The paper is devoted mainly to the corresponding ideals of nowhere dense sets and starts by establishing some (non-)inclusions. The ideals are tall and \(F_{\sigma\delta}\)-subsets of the Cantor set but not \(F_\sigma\). None of the ideals can be extended to summable ideals.
scientific article; zbMATH DE number 6946126
scientific article