Sharp Threshold Asymptotics for the Emergence of Additive Bases

TL;DR

This paper investigates the probability threshold at which a randomly chosen subset of integers becomes a 2-additive basis for a large interval, using probabilistic methods to identify sharp transition points.

Contribution

It establishes sharp threshold asymptotics for the emergence of 2-additive bases in random subsets, extending to k-additive bases.

Findings

Identifies a precise threshold probability for 2-additive basis emergence.

Uses Stein-Chen method and Janson's inequalities for analysis.

Provides generalizations to k-additive bases.

Abstract

A subset A of {0,1,...,n} is said to be a 2-additive basis for {1,2,...,n} if each j in {1,2,...,n} can be written as j=x+y, x,y in A, x<=y. If we pick each integer in {0,1,...,n} independently with probability p=p_n tending to 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is [(1-alpha)n,(1+alpha)n] (or equivalently [alpha n, (2-alpha) n]) for some 0<alpha<1. Under either model, the Stein-Chen method of Poisson approximation is used, in conjunction with Janson's inequalities, to tease out a very sharp threshold for the emergence of a 2-additive basis. Generalizations to k-additive bases are then given.