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

For a large prime number p and a set $A\subset F_p$ we prove the following:

• (1)
  If $A(A+A)$ does not cover all nonzero residues in $F_p$, then $\left|A\right|⩽p/8+o\left(p\right)$.
• (2)
  If A is both sum-free and satisfies $A=A^{⁎}$, then $\left|A\right|⩽p/9+o\left(p\right)$.
• (3)
  If $\left|A\right|\gg \frac{\log \log p}{\sqrt{\log p}}p$, then $|A+A^{⁎}|⩾(1+o(1\left)\right)\min (2\sqrt{\left|A\right|p},p)$.

Here the constants 1/8, 1/9, 2 are the best possible. Proofs make use of wrappers, subsets of a finite abelian group G, which ‘wrap’ popular values in convolutions of dense sets $A,B\subseteq G$. These objects carry certain structural features, making them capable of addressing additive-combinatorial and enumerative problems.