Let
$p$ be a large enough
prime number. When
$A$
is a subset of
${\mathbb{F}}_{p}\backslash \left\{0\right\}$
of cardinality
$\leftA\right>\left(p+1\right)\u22153$,
then an application of the Cauchy–Davenport theorem gives
${\mathbb{F}}_{p}\backslash \left\{0\right\}\subset A\left(A+A\right)$.
In this note, we improve on this and we show that
$\leftA\right\ge 0.3051p$ implies
$A\left(A+A\right)\supseteq {\mathbb{F}}_{p}\backslash \left\{0\right\}$.
In the opposite direction we show that there exists a set
$A$ such
that
$\leftA\right>\left(\frac{1}{8}+o\left(1\right)\right)p$
and
${\mathbb{F}}_{p}\backslash \left\{0\right\}\u2288A\left(A+A\right)$.
