I read in an article of Erdős ("Extremal problems in number theory") that he had a proof of the multiplicative version of the Erdős-Turán conjecture. The statement of this theorem is

Let $a_1 < a_2 < \cdots$ and denote by $g(n)$ the number of solutions to $n=a_ia_j$. Then $g(n)>0$ for all $n>n_0$ implies $$\limsup_{n\to \infty} \ g(n)=\infty.$$

He also claims that this follows under the weaker assumption: let $A(x)=\sum_{a_i < x} 1$. Assume that for every $k$ we have $$\limsup_{x\to \infty} \ A(x)\left(x\left(\frac{\log\log x}{\log x}\right) ^k \right)^{-1}=\infty$$ than the same conclusion follows. However he also says that the proofs are difficult and haven't been published yet.

Since that article is from 1965 I am assuming he must have published something about this theorem afterward, but I don't have a reference. I have seen a proof of the first statement before (not by Erdős), but not the second one. Does anyone know if these proofs were published, simplified or generalized?

I read in an article of Erdős ("Extremal problems in number theory") that he had a proof of the multiplicative version of the Erdős-Turán conjecture. The statement of this theorem is

Let $a_1 < a_2 < \cdots$ and denote by $g(n)$ the number of solutions to $n=a_ia_j$. Then $g(n)>0$ for all $n>n_0$ implies $$\limsup_{n\to \infty} \ g(n)=\infty.$$

He also claims that this follows under the weaker assumption: let $A(x)=\sum_{a_i < x} 1$. Assume that for every $k$ we have $$\limsup_{x\to \infty} \ A(x)\left(x\left(\frac{\log\log x}{\log x}\right) ^k \right)^{-1}=\infty$$ than the same conclusion follows. However he also says that the proofs are difficult and haven't been published yet.

Since that article is from 1965 I am assuming he must have published something about this theorem afterward, but I don't have a reference. I have seen a proof of the first statement before (not by Erdős), but not the second one. Does anyone know if these proofs were published, simplified or generalized?