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For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove thefollowing:  $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then$|A| < p/8 + o(p)$.  $(ii)$ If $A$ is both sum-free and satisfies $A = A^*$, then $|A| < p/9 +o(p)$.  $(iii)$ If $|A| \gg \frac{\log\log{p}}{\sqrt{\log{p}}}p$, then $|A + A^*|\geqslant (1 - o(1))\min(2\sqrt{|A|p}, p)$.  Here the constants $1/8$, $1/9$, and $2$ are the best possible. The proofinvolves \emph{wrappers}, subsets of a finite abelian group $G$, with which we`wrap' popular values in convolutions $A * B$ for dense sets $A, B \subseteqG$. These objects carry some special structural features, making them capableof addressing both additive-combinatorial and enumerative problems.

A new bound for $A(A + A)$ for large sets