Permutations preserving sums of rearranged real series (Q352747)

A permutation \(p: \mathbb N \to \mathbb N\) is said to be preserving sums of series if, for every convergent series \(\sum_{n=1}^\infty a_n\) of reals, the sum of the rearranged series \(\sum_{n=1}^\infty a_{p(n)}\) equals \(\sum_{n=1}^\infty a_n\) whenever \(\sum_{n=1}^\infty a_{p(n)}\) is convergent. The author introduces a class \(\mathfrak S_0\) of those permutations \(p: \mathbb N \to \mathbb N\) for which there is an integer \(k = k(p)\) with the following property: For every \(n \in \mathbb N\) there are finite sets \(A_n, B_n \subset \mathbb N\) such that \(p(A_n) = B_n\), \(\{1, \ldots, n\} \subset A_n\) and each of \(A_n, B_n\) is a union of at most \(k\) mutually disjoint intervals of naturals. The author studies basic properties of \(\mathfrak S_0\) and states the hypothesis that \(\mathfrak S_0\) coincides with the class of permutations preserving sums of series. For the motivation and related results see, for example [\textit{A. Kronrod}, Mat. Sb., N. Ser. 18(60), 237--280 (1946; Zbl 0061.11809); \textit{P. Schaefer}, Am. Math. Mon. 88, 33--40 (1981; Zbl 0455.40007); \textit{C. St. J. A. Nash-Williams} and \textit{D. J. White}, J. Lond. Math. Soc., II. Ser. 59, No. 2, 637--646 (1999; Zbl 0922.40004)].