Achievement sets of series in $\mathbb{R}^2$

Abstract

We examine properties of achievement sets of series in $\mathbb{R}^2$. We show several examples of unusual sets of subsums on the plane. We prove that we can obtain any set of P-sums as a cut of an achievement set in $\mathbb{R}^2.$ We introduce a notion of the spectre of a set in an Abelian group, which generalizes the notion of the center of distances. We examine properties of the spectre and we use it, for example, to show that the Sierpi\'nski carpet is not an achievement set of any series.