On Sidon sets which are asymptotic bases of order $4$

Abstract

Let $h \geq 2$ be an integer. We say that a set $\mathcal{A}$ of positive integers is an asymptotic basis of order $h$ if every large enough positive integer can be represented as the sum of $h$ terms from $\mathcal{A}$. A set of positive integers $\mathcal{A}$ is called a Sidon set if all the sums $a+b$ with $a,b \in \mathcal{A}$, $a \leq b$ are distinct. In this paper we prove the existence of Sidon set $\mathcal{A}$ which is an asymptotic basis of order $4$ by using probabilistic methods.