Not sure whether @David is still around here, but I'd like to add a complement to @quid's answer.

Fix an integer $n \ge 9$, and let $q$ be a prime power and $\mathbb G = (G, \cdot)$ the projective special linear group ${\rm PSL}_n(q)$. It is known, see [4, Section 4], that there exist $a,b \in G$ such that $a^2 = b^{n-1} = 1$ and $G = \langle a, b \rangle$.

Let $I, J \subseteq [\![0,n-2]\!]$ be such that $|I| + |J| > n$. It then follows from an old theorem from the folklore of additive theory, see e.g. [1, Lemma 3.1.2], that $I+J$ covers all residue classes modulo $n-1$.

Therefore, if $X := \{ab^i: i \in I\}$ and $Y := \{b^j: j \in J\}$, then $XY = a\;\!\langle b \rangle$, viz. $|XY| = n-1$, while $|X|+|Y|-1=|I|+|J|-1 > n-1$ and $|G| = \frac{1}{q-1}\prod_{i=0}^{n-1} (q^n-q^i)$.

This is sensibly different from quid's construction, in that $X$ and $Y$ are, roughly speaking, far from being "smooth sets" (in quid's answer, cosets of a subgroup of $\mathbb G$). In addition, we are now *really* working in $\mathbb G$, in the sense that $\langle X, Y \rangle = G$.

Again, the above shows that a *global* analogue of the (classical) Cauchy-Davenport theorem for a group $\mathbb G = (G, \cdot)$, either simple or not, pretending that $|XY| \ge \min(|G|,|X|+|Y|-1)$ for all $X,Y \subseteq G$, is pretty much optimistic.

The Hamidoune-Károlyi theorem, referred to by quid in his answer (see Note (i) below), is a possibility, but it gives a bound that is often too pessimistic.

An alternative is then to generalize the Cauchy-Davenport theorem by rather taking into account the *local* behavior of the ambient group, which can be even done in the broader setting of semigroups.

First, some notation. Suppose $\mathbb A = (A, +)$ is an additively written (but not necessarily commutative) semigroup, that is a set with a binary associative operation on it; we denote by $\mathbb A^\times$ the set of units (or invertible elements) of $\mathbb A$, so that $\mathbb A^\times \ne \emptyset$ iff $\mathbb A$ is a monoid (viz., a unital semigroup).

Given $X \subseteq A$, we define the Cauchy-Davenport constant of $X$ (relative to $\mathbb A$) by $$\gamma(X) := \sup_{x_0 \in X^\times} \inf_{x_0 \ne x \in X} {\rm ord}(x-x_0),$$ where $X^\times := X \cap \mathbb A^\times$, $\inf(\emptyset) := \infty$, $\sup(\emptyset) := 0$, and for an element $z \in A$ we let ${\rm ord}(z)$ be the order of $z$ in $\mathbb A$, namely the cardinality of the subsemigroup of $\mathbb A$ generated by $z$.

Then, for a nonempty $n$-tuple $(X_1, \ldots, X_n)$ of subsets of $A$, we put $$\gamma(X_1, \ldots, X_n) := \max(\gamma(X_1), \ldots, \gamma(X_n)),$$ and call $\gamma(X_1, \ldots, X_n)$ as the Cauchy-Davenport constant of the $n$-tuple $(X_1, \ldots, X_n)$.

With these definitions in mind, it is then possible to prove the following:

**Theorem 1.** If $\mathbb A$ is a cancellative semigroup, then $|X+Y| \ge \min(\gamma(X+Y), |X|+|Y|-1)$ for all $X,Y \subseteq A$.

Here, the sumset of two subsets of $A$ is defined in the very same way as in the context of groups, and $\mathbb A$ being cancellative means that the functions $A \to A: x \mapsto a+x$ and $A \to A: x \mapsto x+a$ are both injective for all $a \in A$.

Moreover, Theorem 1 can be strengthened to the following:

**Theorem 2.** If $\mathbb A$ is a cancellative semigroup, then $|X+Y| \ge \min(\gamma(X, Y), |X|+|Y|-1)$ for all $X,Y \subseteq A$ such that at least one of $X$ and $Y$ generates a commutative subsemigroup of $\mathbb A$.

This is really a strengthening of Theorem 1, as we have:

**Lemma.** If $\mathbb A$ is a cancellative semigroup, then for all $X,Y \subseteq A$, it holds $$\gamma(X,Y) \ge \gamma(X+Y) \ge \mathfrak{p}(\mathbb A),$$ where $\mathfrak{p}(\mathbb A)$ is the infimum of $|S|$ as $S$ ranges over the nontrivial sub*groups* of $\mathbb A$.

Of course, Theorems 1 and 2 are trivial when $\mathbb A^\times = \emptyset$, but this is not the case, e.g., when $\mathbb A$ is a group, and the lemma above shows that both of them are stronger than, and a generalization of, the Hamidoune-Károlyi theorem.

**Added later.** It is perhaps worth mentioning that, if $\mathbb G$, $X$, $Y$, $I$, $J$ and $n$ are given as in the first part of this answer, then the bound implied by the Hamidoune-Károlyi theorem is $|XY| \ge 2$, whereas for $n$ prime (I don't know about other cases) the bound implied by Theorem 2 is $|XY| \ge n-1$, because $\langle Y \rangle$ is a commutative subgroup of $\mathbb G$ and at least one between the sets $I$ and $J$ has size (strictly) larger than $\frac{1}{2}(n-1)$, so that [1, Lemma 3.1.2] applies again and yields $\gamma(X, Y) = n-1$. To wit, Theorem 2 gives the correct size of $XY$, which is comparatively much larger than the bound derived from the Hamidoune-Károlyi theorem if $n \gg 2$, and explains, I hope, the reason why I claimed that the Hamidoune-Károlyi theorem is often too pessimsitic.

**Notes.**

(i) The result is more or less straightforward in the commutative case (by Kneser's theorem), and G. Károlyi proved it for finite groups in 2005 (based on the Feit-Thompson theorem). A proof of the general statement (relying on the isoperimetric method) was then communicated by H. O. Hamidoune to Károlyi during the peer-review process of [2], where it was finally included, see [2, p. 242]. However, Károlyi himself pointed out in a private communication, as recently as July 2013, that a simpler approach comes from a Kneser-type result of J. E. Olson [3, Theorem 2], based on Kemperman's transform. And another argument along the same lines was mentioned by I. Ruzsa in a private communication in June 2013.

(ii) I'm not sure if I should provide a reference for this stuff, as I know that self-advertising is unfair. In any case, the material can be perhaps of interest to some peps here, and seems relevant to the topic, which is why I took the freedom to post it.

**References.**

[1] Y. O. Hamidoune, "The isoperimetric method", in: A. Geroldinger and I. Z. Ruzsa, *Combinatorial Number Theory and Additive Group Theory* (Birkhäuser, 2009), 87-210.

[2] G. Károlyi, *The Cauchy-Davenport theorem in group
extensions*, Enseign. Math. **51** (2005) 239-254.

[3] J. E. Olson, *On the Sum of Two Sets in a Group*, J. Number Th. **18** (1984), 110-120.

[4] M. C. Tamburini and J. S. Wilson, *On the Generation of Finite Simple Groups by Pairs of Subgroup*, J. Algebra **116** (1988), 316-333.