Finite Representability of Integers as $2$-Sums

A set $mathcal{A}$ is said to be an additive $h$-basis if each element in ${0,1,ldots,hn}$ 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 $jin[alpha n, (2-alpha)n]$ 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.