# 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 $XXX=\langle{X\rangle}$?
– C.S.
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 $X$. Aug 6 '18 at 13:52
• Does anyone have the results for small values of $n$? Aug 21 '18 at 8:57
• I expect that its pretty doable for 1 through 4. But then, for 5 it seems quite hard. Aug 23 '18 at 1:27

I would guess that the answer is $$O(n!/n)$$. You can get an example of this size as follows. Partition $$\Omega = \{1, \dots, n\}$$ as $$\{1\} \cup T \cup S$$ where $$|T| = t$$, and let $$X$$ be the set of all $$\pi$$ such that $$\pi(1), \pi^{-1}(1) \in T$$ and $$\pi(T) \subset S$$. The density of $$X$$ is then comparable to $$(t/n)^2 (1 - t/n)^t \approx (t/n)^2 \exp(-t^2/n)$$. The best choice for $$t$$ is $$cn^{1/2}$$ for some constant $$c$$. Meanwhile, by design, $$X^{-1} \cap XX = \emptyset$$, so $$XXX$$ does not contain $$1$$.
More is known about a slight variant of your question. Suppose you want to know the minimal density $$\alpha$$ such that if $$X, Y, Z$$ each have density at least $$\alpha$$ then $$XYZ = G$$. Let $$G = A_n$$. Gowers's method gives the upper bound $$n^{-1/3}$$. An example like the one above gives a lower bound $$n^{-1/2}$$. The truth is $$n^{-1/2+o(1)}$$. I wrote a paper about this a few years ago, which you can find here: https://arxiv.org/abs/1512.03517.