Spectre operator, achievement sets and sets of P-sums in a hyperspace of compact sets
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| Format: | Preprint |
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2025
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| _version_ | 1866914199521722368 |
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| author | Nowakowski, Piotr Prus-Wiśniowski, Franciszek Turoboś, Filip |
| author_facet | Nowakowski, Piotr Prus-Wiśniowski, Franciszek Turoboś, Filip |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_11803 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectre operator, achievement sets and sets of P-sums in a hyperspace of compact sets Nowakowski, Piotr Prus-Wiśniowski, Franciszek Turoboś, Filip General Topology Functional Analysis Group Theory 20K45, 26E25, 11B99 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. |
| title | Spectre operator, achievement sets and sets of P-sums in a hyperspace of compact sets |
| topic | General Topology Functional Analysis Group Theory 20K45, 26E25, 11B99 |
| url | https://arxiv.org/abs/2512.11803 |