On the Erd{o}s-Tur\'an Conjecture and the growth of B₂[g] sequences

When g we say that A is a B₂[g] sequence if every m has at most g distinct representations of the shape m=b₁+b₂ with b₁≤ b₂ and b₁,b₂∈ A. We show for every 0<ε<1 that whenever g>1/ε then there is a B₂[g] sequence A having the property that every sufficiently large n can be written as n=a₁+a₂+a₃,\ \ \ \ \ \ \ \ \ a₃≤ nε\ \ \ \ \ \ \ \ \ aᵢ∈ A, and satisfying for large x the estimate A∩ [1,x] xg/(2g+1). The above lower bound improves upon earlier results of Cilleruelo and of Erd{o}s and Renyi.