A subset $A$ of $\mathbb{N}$ is called a basis of order $k$ if the set $kA$ = {$a_1 + \cdots + a_k | a_1, \cdots, a_k \in A$} $= \mathbb{N} \setminus C$, where $C$ is a finite set of positive integers. Let $r_{k,A}(N)$ be the number of representations of $N$ as a sum of $k$ (not necessarily distinct) elements of $A$. Erdos proved the following theorem in 1956:

There exists a basis $B \subset \mathbb{N}$ such that $r_{2,B}(N) = \Theta (\log(n))$ for every sufficiently large $n$.

In 1990, together with Tetali, he proved the following generalization.

For any fixed $k \in \mathbb{N}$, there exists a basis $B \subset \mathbb{N}$ such that $r_{k,B}(N) = \Theta (\log(n))$.

Now my question is, can there be any 'thinner' bases? That is, can we improve on the $\log(n)$ in the above theorems? Erdos and Turan conjectured that $\limsup_{n \rightarrow \infty} r_{2,B}(n) = \infty$ and later Erdos conjectured that $\limsup_{n \rightarrow \infty} r_{2,B}(n)/\log(n) > 0$ for any basis $B$ of order 2, suggesting that the answer is that one cannot really improve on the $\log(n)$. Are there are progress in this direction, either positive or negative?