$\omega \subseteq \mathbb{R}^+$ is called a Sidon sequence,
if all the sums $a + a' \ (a, a' \in \omega, a \leq a')$ are distinct,
and it is an asymototic basis of order $2$,
if any positive integer $n$ sufficiently large can be expressed as a sum of $2$ elements of $\omega$.


According to the article I am reading, apparently it is not too difficult to show that there does not exist $\omega$ such that it is a Sidon sequence and also an asymototic basis of order $2$, but I am not quite seeing how to prove this at the moment. 


I was wondering if someone could possibly give me an explanation on how to show this? Thank you very much!



No Sidon sequence which is an asymptotic basis of order $2$
Sidon set $A$ has at most $\sqrt{n}(1+o(1))$ elements not exceeding $n$ (\*). So, $A+A$ contains at most $|A|(|A|+1)/2=n(1/2+o(1))$ elements not exceeding $n$, unlike an asymptotic basis of order 2. 


(\*) may be proved as follows: fix $M$, denote your elements $x\_1<x\_2<\dots<x\_m\leqslant n$ and consider all differences $x\_j-x\_i$ for $i<j\leqslant i+M$. They are all distinct, there areat least, say, $M(m-M)$ such differences and the sum of them does not exceed $\frac{M(M+1)}2 n$, thus $\frac{M(M+1)}2 n\geqslant (M(m-M))^2/2$, optimizing by $M$ or just by taking very large $M$ we get what we need: $m\leqslant \sqrt{n}(1+o(1))$.


