There's a fairly simple result that states that
for a finite group $G$ and two subsets $A, B$ with $|A| + |B| > |G|,$ any $g \in G$ has a representation $g = a*b$ with $a \in A$, $b \in B$.
To prove this, just consider that the sets $A$ and $gB^{-1} = \{gb^{-1} : b \in B\}$ can't be disjoint.
This immediately implies that every number modulo a prime $p$ is the sum of two squares:
let $G = \mathbb{Z}/p\mathbb{Z}$ and $A = B = \{x^2 : x \in \mathbb{Z}/p\mathbb{Z}\}$.
My question: can this result be extended meaningfully to products of three or more subsets? I'm interested in representations mod $p$ as sums of cubes and higher powers.
The most obvious (to me) generalization
for a finite group $G$ and three subsets $A, B, C$ with $|A| + |B| + |C| > |G|,$ any $g \in G$ has a representation $g = a*b*c$ with $a \in A$, $b \in B$, $c \in C$.
is clearly false: just take $G = \mathbb{Z}/2\mathbb{Z}$ and $A = B = C = 0$.