# Finite Representability of Integers as $2$-Sums

## TL;DR

This paper investigates the properties of randomly selected sets as truncated additive bases, providing precise asymptotic results for their likelihood of representing integers in a specified range as sums of two elements.

## Contribution

It introduces the concept of truncated $(eta,2,g)$ additive bases and derives sharp asymptotic probabilities for random sets to be such bases.

## Key findings

- Sharp asymptotics for the probability of random sets being truncated additive bases.
- Characterization of the number of representations of integers as 2-sums within a range.
- Analysis of high and low probability regimes for these bases.

## Abstract

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 $j\in[\alpha n, (2-\alpha)n]$ 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.