On the independence number of sparser random Cayley graphs

The Cayley sum graph ${\Gamma }_A$ of a set $A\subseteq Z_n$ is defined to have vertex set $Z_n$ and an edge between two distinct vertices $x,y\in Z_n$ if $x+y\in A$. Green and Morris proved that if the set $A$ is a $p$-random subset of $Z_n$ with $p=1/2$, then the independence number of ${\Gamma }_A$ is asymptotically equal to $\alpha \left(G\right(n,1/2\left)\right)$ with high probability. Our main theorem is the first extension of their result to $p=o\left(1\right)$: we show that, with high probability,

One of the tools in our proof is a geometric-flavoured theorem that generalises Freĭman's lemma, the classical lower bound on the size of high-dimensional sumsets. We also give a short proof of this result up to a constant factor; this version yields a much simpler proof of our main theorem at the expense of a worse constant.