An explicit economical additive basis

We present an explicit subset $A\subseteq \mathbb{N}=\{0,1,\dots \}$ such that $A+A=\mathbb{N}$ and for all $\epsilon >0$, $$\underset{N\to \mathrm{\infty }}{\lim }\frac{\mid \{(n_1,n_2):n_1+n_2=N,(n_1,n_2)\in A^2\}\mid }{N^{\epsilon }}=0.$$ This answers a question of Erdős.