# What is $\tau(A_n)$?

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 $\tau(A_n)$?

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...
• I don't even understand the question. What do you mean when you say $XXX=\langle{X\rangle}$? – crskhr Aug 6 '18 at 13:44
• @crskhr, $XXX$ stands for the group subset product: $XXX = \{abc| a, b, c \in X\}$; $\langle X \rangle$ stands for group subset closure: $\langle X \rangle$ is the minimal subgroup that contains $X$. – Yanior Weg Aug 6 '18 at 13:52
• Does anyone have the results for small values of $n$? – mathworker21 Aug 21 '18 at 8:57