Bounded ranges of cardinal functions

TL;DR

This paper investigates the possible value ranges of cardinal functions associated with sequences of positive real numbers, providing constructions, criteria, and a complete characterization for certain cases, especially when the maximum element is 6.

Contribution

It offers new criteria and constructions for the ranges of cardinal functions and fully characterizes these ranges for interval-filling sequences with maximum element 6.

Findings

Characterization of ranges for sequences with maximum element 6

Development of criteria for sets to be ranges of cardinal functions

Construction methods for sequences with prescribed cardinal function ranges

Abstract

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 $f:\mathcal{A}(\mathbf{x}) \to \mathbb{N}\cup\{\omega,\mathfrak{c}\}$ 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.