  Let $p$ be a prime, let $S$ be a non-empty subset of $\mathbb{F}_p$ and let
$0<\epsilon\leq 1$. We show that there exists a constant $C=C(p, \epsilon)$
such that for every positive integer $k$, whenever $\phi_1, \dots, \phi_k:
\mathbb{F}_p^n \rightarrow \mathbb{F}_p$ are linear forms and $E_1, \dots, E_k$
are subsets of $\mathbb{F}_p$, there exist linear forms $\psi_1, \dots, \psi_C:
\mathbb{F}_p^n \rightarrow \mathbb{F}_p$ and subsets $F_1, \dots, F_C$ of
$\mathbb{F}_p$ such that the set $U=\{x \in S^n: \psi_1(x) \in F_1, \dots,
\psi_C(x) \in F_C\}$ is contained inside the set $V=\{x \in S^n: \phi_1(x) \in
E_1, \dots, \phi_k(x) \in E_k\}$, and the difference $V \setminus U$ has
density at most $\epsilon$ inside $S^n$. We then generalize this result to one
where $\phi_1, \dots, \phi_k$ are replaced by homomorphisms $G^n \to H$ for
some pair of finite Abelian groups $G$ and $H$, and to another where they are
replaced by polynomial maps $\mathbb{F}_p^n \to \mathbb{F}_p$ of small degree.
