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

2025/11/02 by Piotr Nowakowski, Franciszek Prus‐Wiśniowski, Nowakowski, Piotr +4 · 1 voice · 2 citations
Mathematics · #11B99 #20K45 #26E25 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Group Theory (math.GR) #Limits and Structures in Graph Theory #math.FA #math.GN #math.GR

paper · pdf · doi:10.48550/arxiv.2512.11803

openalex publication_date 2025/11/02 · arxiv published 2025/11/02 · arxiv updated 2025/11/02 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28

Abstract

Let (X,+,d) be an Abelian metric group and A⊂ X. We investigate the spectre of a set A, defined as the set of all elements z∈ X such that for every x∈ A either x+z ∈ A or x-z ∈ 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.

Citations

• Center of distances and Bernstein sets
• the center of distances for some multigeometric series
• On the center of distances

Cited by

• Achievement sets -- current results and open problems
• Center of distances of ultrametric spaces generated by labeled trees

Discussions

• Piotr Nowakowski, Franciszek Prus-Wi\'sniowski, Filip Turobo\'s: Spectre operator, achievement sets and sets of P-sums in a hyperspace of compact sets https://arxiv.org/abs/2512.11803 https://arxiv.or [bsky, 0 points, 0 comments]

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