There seems to be some combinatorial fact that every subset $A$ of  $G=(\mathbb{Z}/p)^{\times n}$ of cardinality $\frac{p^n-1}{p-1}+1$ containing $\vec{0}$ satisfies $(p-1)A=G$. ($p$ is a prime number.)
Is that true and known in the literature?
I would appreciate a reference or a proof.
Thanks!