  Let $\Gamma < G$ be a discrete subgroup of a locally compact unimodular group
$G$. Let $m\in C_b(G)$ be a $p$-multiplier on $G$ with $1 \leq p < \infty$ and
let $T_{m}: L_p(\widehat{G}) \rightarrow L_p(\widehat{G})$ be the corresponding
Fourier multiplier. Similarly, let $T_{m \vert_\Gamma}: L_p(\widehat{\Gamma})
\rightarrow L_p(\widehat{\Gamma})$ be the Fourier multiplier associated to the
restriction $m|_{\Gamma}$ of $m$ to $\Gamma$. We show that
  \[
  c( {\rm supp}( m\vert_{\Gamma} ) ) \Vert T_{m \vert_\Gamma}:
L_p(\widehat{\Gamma}) \rightarrow L_p(\widehat{\Gamma}) \Vert \leq \Vert T_{m
}: L_p(\widehat{G}) \rightarrow L_p(\widehat{G}) \Vert, \] for a specific
constant $0 \leq c(U) \leq 1$ that is defined for every $U \subseteq \Gamma$.
  The function $c$ quantifies the failure of $G$ to admit small almost
$\Gamma$-invariant neighbourhoods and can be determined explicitly in concrete
cases. In particular, $c(\Gamma) =1$ when $G$ has small almost
$\Gamma$-invariant neighbourhoods. Our result thus extends the De Leeuw
restriction theorem from [CPPR15] as well as De Leeuw's classical theorem
[Lee65].
  For real reductive Lie groups $G$ we provide an explicit lower bound for $c$
in terms of the maximal dimension $d$ of a nilpotent orbit in the adjoint
representation. We show that $c(B_\rho^G) \geq \rho^{-d/4}$ where $B_\rho^G$ is
the ball of $g\in G$ with $\Vert {\rm Ad}_g \Vert < \rho$.
  We further prove several results for multilinear Fourier multipliers. Most
significantly, we prove a multilinear De Leeuw restriction theorem for pairs
$\Gamma<G$ with $c(\Gamma) = 1$. We also obtain multilinear versions of the
lattice approximation theorem, the compactification theorem and the
periodization theorem. Consequently, we are able to provide the first examples
of bilinear multipliers on nonabelian groups.
