Achievement sets and sum ranges with ideal supports

Jacek Marchwicki

We introduce the notion of ideally supported achievement sets for a series of real numbers. We analyze 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 Filipow and Szuca. We complete Filip ´ow and Szuca characterization of ideal sum ranges, [R. Filipow, 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.