Weighted real Egyptian numbers
Melvyn B. Nathanson

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 be a sequence of nonempty finite sets of positive real numbers, and let be a sequence of infinite discrete sets of positive real numbers. A weighted real Egyptian number with numerators and denominators is a real number that can be represented in the form \[ c = \sum_{i=1}^n \frac{a_i}{b_i} \] with and for . In this paper, classical results of Sierpinski for Egyptian fractions are extended to the set of weighted real Egyptian numbers.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Computability, Logic, AI Algorithms · semigroups and automata theory
