Dense sumsets of Sidon sequences

TL;DR

This paper proves the existence of a Sidon set with a positive lower density of its three-fold sumset, using probabilistic methods, addressing a long-standing question about Sidon sets as asymptotic bases.

Contribution

It demonstrates the existence of a Sidon set whose triple sumset has positive lower density, solving a problem posed by Erdős, Sárközy, and Sós.

Findings

Existence of Sidon set with positive lower density of A+A+A

Use of probabilistic methods to construct such sets

Addresses a long-standing open problem in additive number theory

Abstract

Let $k \ge 2$ be an integer. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called Sidon set if all the two terms sums formed by the elements of $A$ are different. Many years ago P. Erd\H{o}s, A. S\'ark\"ozy and V. T. S\'os asked whether there exists a Sidon set which is asymptotic basis of order $3$. In this paper we prove the existence of a Sidon set $A$ with positive lower density of the three fold sumset $A + A + A$ by using probabilistic methods.