For any set $B\subseteq \mathbb{N}$ one can associate the formal series $$f_B(z) = \sum_{b\in B}z^b$$ and obtain $$f_B(z)^k = \sum_{n\geqslant 0} r_{B,k}(n)z^n,$$ where $r_{B,k}(n) = |\{(x_1,\cdots,x_k)\in B^k : x_1+\cdots+x_k=n\}|$. As it is characteristic of the Circle Method, one them have $$ r_{B,k}(n) = \frac{1}{2\pi i}\int_{|z|=\rho} \frac{f_B(z)^k}{z^{n+1}}~\mathrm{d}z = \int_0^1 f_B(e(t))^k e(-nt)~\mathrm{d}t,$$ where $e(z) = e^{2\pi iz}$.

I am starting to study the Circle Method and the first thing I noticed is that, although it is a very powerful way to deal with both Waring's and weak Goldbach's problems, it seems that it relies heavily on the number-theoretical properties of the primes and Waring bases to work properly, and doesn't have strong "combinatorial powers" *per se*.

However, this first step of the setup seems to be a very very natural starting point to deal with problems involving (asymptotic) bases in general, but I found way less discussions about this in the literature than I expected. More especifically, my question is:

Is there any approach of the Erdős–Turán conjecture on additive bases via the Hardy-Littlewood circle method? Or it is known to be an ineffective approach?

As far as I know, the best results up to date has been made in the closely related problem of finding thin bases, namely

**Erdös-Tetali theorem**(1990) on the existence of asymptotic $h$-bases $B$ with $r_{B,h}(n) = \Theta(\log(n))$, using, basically, only probabilistic methods.**Vu's thin subbases of Waring bases**(2000), showing the existence of asymptotic $h_k$-bases $B^{(k)} \subseteq P_k = \{n^k : n\geqslant 0\}$ with $r_{B^{(k)},h_k}(n) = \Theta(\log(n))$ for every $k$. He uses both probabilistic methods AND the Circle Method, but as far as I understood the last one is used "only" in dealing with the $P_{k}$'s. (By "only" I mean that it is not in the sense that I am asking in this question.)

Other approaches like the Borwein-Choi-Chu result (2006) that asserts that if $B$ is a $2$-basis then $\liminf\limits_{n\to\infty} r_{B,2}(n) > 7$ seems fruitful too, but I didn't spent enough time reading it to be able to make a proper remark. On the other hand, the current probabilistic methods apparently have a "technical limitation" involving the $\Theta(\log(n))$, as it seems to be the best one can get in this direction.

After reading Xiao's master's dissertation and a few books (Alon & Spencer's *Probabilistic Method*, Nathanson's *Additive NT: Classical Bases*, Tao & Vu's *Additive Combinatorics*), I had a feeling that the Circle Method is a possible way to understand better this limitation of the probabilistic method.

In any case, this is more a raw idea than anything. I'd appreciate any references, thoughts and thinghs to think about. Thanks in advance!