  We show that any connected Cayley graph $\Gamma$ on an Abelian group of order
$2n$ and degree $\tilde{\Omega}(\log n)$ has at most $2^{n+1}(1 + o(1))$
independent sets. This bound is tight up to to the $o(1)$ term when $\Gamma$ is
bipartite. Our proof is based on Sapozhenko's graph container method and uses
the Pl\"{u}nnecke-Rusza-Petridis inequality from additive combinatorics.
