On the Erdős-Turán Conjecture and the growth of $B_2[g]$ sequences

When $g\in \mathbb{N}$ we say that $A\subset \mathbb{N}$ is a $B_2[g]$ sequence if every $m\in \mathbb{N}$ has at most $g$ distinct representations of the shape $m=b_1+b_2$ with $b_1\le b_2$ and $b_1,b_2\in A$. We show for every $0<\epsilon <1$ that whenever $g>\frac{1}{\epsilon }$ then there is a $B_2[g]$ sequence $A$ having the property that every sufficiently large $n\in \mathbb{N}$ can be written as $$n=a_1+a_2+a_3,a_3\le n^{\epsilon }{a}_i\in A,$$ and satisfying for large $x$ the estimate $$\mid A\cap [1,x]\mid \gg x^{g\mathrm{/}(2g+1)}\mathrm{.}$$ The above lower bound improves upon earlier results of Cilleruelo and of Erdős and Renyi.