Enumerating independent sets in Abelian Cayley graphs

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 Sapozhenkos graph container method and uses the Pl{u}nnecke-Rusza-Petridis inequality from additive combinatorics.