Weighted real Egyptian numbers

Abstract

Let $\mca = (A_1,\ldots, A_n)$ be a sequence of nonempty finite sets of positive real numbers, and let $\mcb = (B_1,\ldots, B_n)$ be a sequence of infinite discrete sets of positive real numbers. A \emph{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 Sierpiński for Egyptian fractions are extended to the set of weighted real Egyptian numbers.