Weighted real Egyptian numbers

TL;DR

This paper extends classical results on Egyptian fractions to a broader class called weighted real Egyptian numbers, involving finite sets of numerators and infinite sets of denominators, and explores their properties.

Contribution

The paper generalizes Sierpinski's classical results to weighted real Egyptian numbers, broadening the understanding of their structure and representation.

Findings

Extension of Sierpinski's results to weighted real Egyptian numbers

Characterization of representations involving finite and infinite sets

Insights into the structure of these generalized Egyptian numbers

Abstract

Let $\mathcal A = (A_1,\ldots, A_n)$ be a sequence of nonempty finite sets of positive real numbers, and let $\mathcal{B} = (B_1,\ldots, B_n)$ be a sequence of infinite discrete sets of positive real numbers. A weighted real Egyptian number with numerators $\mathcal{A}$ and denominators $\mathcal{B}$ is a real number $c$ that can be represented in the form \[ c = \sum_{i=1}^n \frac{a_i}{b_i} \] with $a_i \in A_i$ and $b_i \in B_i$ for $i \in \{1,\ldots, n\}$. In this paper, classical results of Sierpinski for Egyptian fractions are extended to the set of weighted real Egyptian numbers.