Bounded ranges of cardinal functions
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911110223888384 |
|---|---|
| author | Marchwicki, Jacek Żmija, Błażej |
| author_facet | Marchwicki, Jacek Żmija, Błażej |
| contents | Let $\mathbf{x}$ be a (non-empty) sequence of positive real numbers. Its achievement set $\mathcal{\mathbf{x}}$ is the set of all the possible sums of the elements of $\mathbf{x}$. The cardinal function of $\mathbf{x}$ is the function $f:\mathcal{A}(\mathbf{x}) \to \mathbb{N}\cup\{ω,\mathfrak{c}\}$ that for every $x\in\mathbb{A}(\mathbf{x})$ the value $f(x)$ is equal to the number of ways $x$ is represented as a sum of elements of $\mathbf{x}$.
In this paper we consider possible ranges of cardinal functions of sequences $\mathbf{x}$. We present some general constructions and several criteria that a set has to satisfy in order to be a range of a cardinal function.
We put special attention to the case of sets with maximal element equal to $6$. In this case, in particular, we obtained a full characterisation of sets that are ranges of cardinal functions of interval-filling sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13016 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded ranges of cardinal functions Marchwicki, Jacek Żmija, Błażej Classical Analysis and ODEs Combinatorics 40A05 (Primary) 11K31 (Secondary) Let $\mathbf{x}$ be a (non-empty) sequence of positive real numbers. Its achievement set $\mathcal{\mathbf{x}}$ is the set of all the possible sums of the elements of $\mathbf{x}$. The cardinal function of $\mathbf{x}$ is the function $f:\mathcal{A}(\mathbf{x}) \to \mathbb{N}\cup\{ω,\mathfrak{c}\}$ that for every $x\in\mathbb{A}(\mathbf{x})$ the value $f(x)$ is equal to the number of ways $x$ is represented as a sum of elements of $\mathbf{x}$. In this paper we consider possible ranges of cardinal functions of sequences $\mathbf{x}$. We present some general constructions and several criteria that a set has to satisfy in order to be a range of a cardinal function. We put special attention to the case of sets with maximal element equal to $6$. In this case, in particular, we obtained a full characterisation of sets that are ranges of cardinal functions of interval-filling sequences. |
| title | Bounded ranges of cardinal functions |
| topic | Classical Analysis and ODEs Combinatorics 40A05 (Primary) 11K31 (Secondary) |
| url | https://arxiv.org/abs/2508.13016 |