On the Erdős-Turán Conjecture and the growth of $B_{2}[g]$ sequences

Abstract: 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 $b_{1}\leq b_{2}$ and $b_{1},b_{2}\in A$. We show for every $0<\varepsilon<1$ that whenever $g>\frac{1}{\varepsilon}$ then there is a $B_{2}[g]$ sequence $A$ having the property that every sufficiently large $n\in\mathbb{N}$ can be written as $$n=a_{1}+a_{2}+a_{3},\ \ \ \ \ \ \ \ \ a_{3}\leq n^{\varepsilon}\ \ \ \ \ \ \ \ \ a_{i}\in A,$$ and satisfying for large $x$ the estimate $$\lvert A\cap [1,x]\rvert\gg x^{g/(2g+1)}.$$ The above lower bound improves upon earlier results of Cilleruelo and of Erd\H{o}s and Renyi.