Suppose G is a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$. What is $\tau(A_n)$?

Similar problems for some different classes of groups are already answered:

1) $\tau(\mathbb{Z}_n) = \lceil \frac{n}{3} \rceil + 1$ (this is a number-theoretic fact proved via arithmetic progressions)

2) Gowers, Nikolov and Pyber proved the fact that $\tau(SL_n(\mathbb{Z}_p)) = 2|SL_n(\mathbb{Z}_p)|^{1-\frac{1}{3(n+1)}}$ (this fact is proved with linear algebra)

However, I have never seen anything like that for $A_n$. It will be interesting to know if there is something...