What conditions would be sufficient for a generalization of Cauchy-Davenport for simple groups? I can see two possible difficulties with a generalization for general groups:


1. The sets could both be part of a subgroup of the group.
2. The sets could both be cosets of a normal subgroup. This is impossible for simple groups.


Are these the only ways Cauchy-Davenport can fail, or are there other ways?


In particular, would it be possible to generalize the proof of Cauchy-Davenport given in <http://arxiv.org/pdf/math/0308286v6.pdf> with a more general version of the uncertainty principle used in that paper?


I haven't used mathoverflow before, so apologies if this question isn't appropriate for this website.



Conditions for an analogue of Cauchy-Davenport for simple groups
First, a slightly tangential comment regarding what I assume you mean by 'Cauchy--Davenport fails';
I include it for reader potentially unfamiliar with it and since there is also a somewhat common other way to generalize it; cf below.


The Cauchy--Davenport Theorem asserts that for $G$ a prime cyclic group of order $p$ one has for nonempty subsets $A, B$  

$$|AB| \ge \min \lbrace |A| + |B| - 1, p \rbrace $$


It seems that you mean with 'Cauchy--Davenport fails' that the inequality does not hold (for certain sets) with $p$ in the displayed formula replaced by the order of the group. This is however not the only thing one could call (the analog of) Cauchy--Davenport. Indeed an other usage is somewhat established: there is a paper by J.P. Wheeler called ['The Cauchy--Davenport Theorem for finite groups'](http://arxiv.org/abs/1202.1816) that asserts the displayed equation for finite groups with the understanding that $p$ is the smallest order of a nonidenty element. This was also obtained by Gy. Karolyi (independently).


Now, to the actual question.
Yes, there are somewhat other types of sets. For example, take $A=aH$ and $B=bH$ with $b$ in the normalizer of $H$ or $A=aH$ and $B=Hb$, for some (nonnormal) subgroup $H$.


However, there are results classifying sets for which the product set is very small. 
See the blog post of Tao ['An elementary noncommutaive Freiman--Kneser Theorem'](http://terrytao.wordpress.com/2009/11/10/an-elementary-non-commutative-freiman-theorem/) 


Containing for example the result (originally due to Freiman, cf Seva's anwer for details): if $|A \cdot A|< 3 |A|/2$ then $S = A \cdot A^{-1}$ is a subgroup of order $|A \cdot A|$ and $A \subset a S$ and $aS =Sa$.


In a more recent blog post (['Hamidoune’s Freiman-Kneser theorem for nonabelian groups'](http://terrytao.wordpress.com/2011/03/12/hamidounes-freiman-kneser-theorem-for-nonabelian-groups/)) Tao also discusses [recent work of the late Hamidoune](http://arxiv.org/abs/1006.5074) answering a question raised in the above mentioned blog post, which sort of classifies subsets for which 'Cauchy--Davenport fails' (in part, but not always normal subgroups arise, so in the simple case there are some reductions); roughly one gets unions of cosets. In that blog post also work of Sanders (unpublished at that time, yet [meanwhile available](http://arxiv.org/abs/1212.0457)) is mentioned related to this problem, which methodologically, using Fourier analytic methods, seems closer than Hamidoune's to the uncertainty principle you mention; yet I do not oversee this in detail. 


