  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.
