On subgraphs of random Cayley sum graphs

Let $G$ be a finite sufficiently large additive group and $A$ be a random subset of $G$ such that any element of $A$ belongs to $A$ with probability $1/2$. Mrazovic showed recently that with the probability tending to one the set $A$ does not contain all sums $B+C$ of subsets of $G$ with $|B|, |C| \ge (\log |G|)^{2+o(1)}$. S. V. Konyagin and I. D. Shkredov improved the last result showing that the bound can be refined to $\log |G| (\log \log |G|)^C$ for a certain absolute $C>0$. Moreover under these conditions just a half of sums of $B+C$ belongs to $A$ and up to ($\log \log |G|)^C$ factors the result is tight.