Weighted real Egyptian numbers

Let $\mathcal{A}=(A_1,\dots ,A_n)$ be a sequence of nonempty finite sets of positive real numbers, and let $\mathcal{B}=(B_1,\dots ,B_n)$ 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.