Spectre operator, achievement sets and sets of P-sums in a hyperspace of compact sets

TL;DR

This paper explores the spectral properties of sets within Abelian metric groups, analyzing an operator on compact sets, and investigates achievement sets and P-sums, including properties in the plane.

Contribution

It introduces and studies the spectre operator on compact sets in Abelian metric groups and examines achievement sets and P-sums, providing new insights into their properties.

Findings

Properties of the spectre operator are characterized.

Achievement sets in the plane are analyzed.

Relations between achievement sets and P-sums are established.

Abstract

Let $(X,+,d)$ be an Abelian metric group and $A\subset X$. We investigate the spectre of a set $A$, defined as the set of all elements $z\in X$ such that for every $x\in A$ either $x+z \in A$ or $x-z \in A$. We consider the corresponding to this notion operator $S$ acting on the hyperspace of compact sets and examine its properties. Furthermore, we study the families of achievement sets and sets of $P$-sums in this hyperspace, as well as prove some properties of achievement sets in the plane.