Achievement sets and sum ranges with ideal supports

TL;DR

This paper introduces and analyzes the concept of ideally supported achievement sets for real series, exploring their complexity, topological properties, and relation to sum ranges, extending classical results like Riemann's theorem.

Contribution

It defines ideal achievement sets, compares them with ideal sum ranges, and completes existing characterizations, providing new generalizations of Riemann's theorem.

Findings

Characterization of ideal achievement sets and their topological properties

Comparison and extension of ideal sum ranges with previous work

Generalization of Riemann's theorem for series with ideal supports

Abstract

We introduce the notion of ideally supported achievement sets for a series of real numbers. We analize their complexity and topological properties. We compare the notion of ideal achievement sets with the notion of ideally supported sum range of real series, considered by Filip\'ow and Szuca. We complete Filip\'ow and Szuca characterization of ideal sum ranges, [R. Filip\'ow, P. Szuca, Rearrangement of conditionally convergent series on a small set, J. Math. Anal. Appl. 362 (2010), no. 1, 64-71.], and we obtain some generalization of Riemann's Theorem.