Sums of finitely many distinct rationals

Abstract

${\cal E}$ denotes the family of all finite nonempty $S\subseteq{\mathbb N}:=\{1,2,\ldots\}$, and ${\cal E}(X):={\cal E}\cap\{S:S\subseteq X\}$ when $X\subseteq{\mathbb N}$. Similarly, ${\cal F}$ denotes the family of all finite nonempty $T\subseteq{\mathbb Q}^+$, and ${\cal F}(Y) := {\cal F}\cap\{T:T\subseteq Y\}$ where ${\mathbb Q}^+$ is the set of all positive rationals and $Y\subseteq{\mathbb Q}^+$. This paper treats the functions $\sigma:{\cal E}\rightarrow{\mathbb Q}^+$ given by $\sigma:S\mapsto\sigma S :=\sum\{1/x:x\in S\}$, the function $\delta:{\cal E}\rightarrow{\mathbb N}$ defined by $\sigma S = \nu S/\delta S$ where the integers $\nu S$ and $\delta S$ are coprime, and the more general function $\Sigma:{\cal F}\rightarrow{\mathbb Q}^+$ where $\Sigma T$ denotes the sum of the elements in $T$ for $T\in{\cal F}$. Theorem 1.1. For each $r\in{\mathbb Q}^+$, there exists an infinite pairwise disjoint subfamily ${\cal H}_r\subseteq{\cal E}$ such that $r=\sigma S$ for all $S\in{\cal H}_r$. Theorem 1.2. Let $X$ be a pairwise coprime set of positive integers. Then $\sigma$ restricted to ${\cal E}(X)$ and $\delta$ restricted to ${\cal E}(X)$ are injective. Also, $\sigma C\in{\mathbb N}$ for $C\in{\cal E}(X)$ only if $C=\{1\}$. Theorem 6.5. There is a set $X$ of positive rational numbers for which $\Sigma:{\cal F}(X)\rightarrow{\mathbb Q}^+$ is a surjection, but for which $1\in X$ and the only $S\in{\cal F}(X)$ with $\Sigma S = 1$ is $S = \{1\}$.