A weak form of the Erdős-Turán conjecture This question is motivated by the answer of Gowers to the question Erdos Conjecture on arithmetic progressions.

Question. (1)-Suppose $A \subset \mathbb{N}$ is such that
  Lim$_n$ $log(n) \cdot |A \cap \{ 1, \dots, n\}|\over{n}$=1. Does $A$ contain arithmetic progressions of arbitrary large finite length?
(2) What if we replace Lim with Limsup?

 A: The answer to both questions is almost certainly yes, but this has not been proven. It remains an open question even for progressions of length 3 (although the current bounds are pretty close).
For progressions of length greater than 3, the best bounds are due to Gowers, which stated in your terms give: if 
$$ \limsup \frac{(\log\log N)^{c_k}\lvert A\cap \{1,\ldots,N\}\rvert}{N}>0,$$
where $c_k>0$ is some constant depending only on $k$, then $A$ contains infinitely many arithmetic progressions of length $k$. For progressions of length 3 the $(\log\log N)^{c_k}$ can be replaced by $\log N(\log\log N)^{-4}$. For progressions of length 4 it becomes $(\log N)^{c}$, which is a recent result of Green and Tao. Obtaining even this kind of bound, let alone answering your question, for progressions of length 5 or longer seems quite far out of each at the moment.
Based on the constructions we know, something like the following is probably true: if 
$$ \limsup \frac{\exp((\log N)^{c_k})\lvert A\cap \{1,\ldots,N\}\rvert}{N}>0$$
then $A$ contains infinitely many arithmetic progressions of length $k$.
