Finite Representability of Integers as $2$-Sums

Abstract

A set $\mathcal{A}$ is said to be an additive $h$-basis if each element in $\{0,1,\dots ,hn\}$ can be written as an $h$-sum of elements of $\mathcal{A}$ in {\it at least} one way. We seek multiple representations as $h$-sums, and, in this paper we make a start by restricting ourselves to $h=2$. We say that $\mathcal{A}$ is said to be a truncated $(\alpha ,2,g)$ additive basis if each $j\in [\alpha n,(2-\alpha )n]$ can be represented as a $2$-sum of elements of $\mathcal{A}$ in at least $g$ ways. In this paper, we provide sharp asymptotics for the event that a randomly selected set is a truncated $(\alpha ,2,g)$ additive basis with high or low probability.

14 pages