Asymptotics for the logarithm of the number of $k$-solution-free sets in Abelian groups

A family $(A_1,\dots,A_k)$ of subsets of a group $G$ is called $k$-solution-free family if the equation $x_1+\dots+x_k=0$ has no solution in $(A_1,\dots,A_k)$ such that $x_1\in A_1,\dots,x_k\in A_k$. We find the asymptotic behavior for the logarithm of the number of $k$-solution-free families in Abelian groups.