On $B_{h}[1]$-sets which are asymptotic bases of order $2h$

TL;DR

This paper proves the existence of special sets of positive integers, called $B_{h}[1]$-sets, that serve as asymptotic bases of order $2h$, using probabilistic techniques to demonstrate their existence.

Contribution

It establishes the existence of $B_{h}[1]$-sets that are asymptotic bases of order $2h$, a novel result in additive number theory.

Findings

Existence of $B_{h}[1]$-sets as asymptotic bases of order $2h$

Use of probabilistic methods to prove set existence

Advancement in understanding the structure of additive bases

Abstract

Let $h,k \ge 2$ be integers. A set $A$ of positive integers is called asymptotic basis of order $k$ if every large enough positive integer can be written as the sum of $k$ terms from $A$. A set of positive integers $A$ is said to be a $B_{h}[g]$-set if every positive integer can be written as the sum of $h$ terms from $A$ at most $g$ different ways. In this paper we prove the existence of $B_{h}[1]$ sets which are asymptotic bases of order $2h$ by using probabilistic methods.