Finite Representability of Integers as $2$-Sums

Abstract

A set A = A k,n ⊂ [n] ∪ {0} is said to be an additive h-basis if each element in {0, 1, . . . , hn} can be written as an h-sum of elements of A in 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 A is said to be a truncated (α, 2, g) additive basis if each j ∈ [αn, (2 − α)n] can be represented as a 2-sum of elements of A k,n in at least g ways. In this paper, we provide sharp asymptotics for the event that a randomly selected set is a truncated (α, 2, g) additive basis with high or low probability.