Let A $\subset [0:2]^n$, where $[0:2]=\{0,1,2\}$, then define $2A= \{ a+b\mid a,b \in A \}$. I wanted to know the best known lower-bound estimates for $|2A|$.

I intuitively expect that $|2A| \geq |A|^{\log_3{5}}$ as this estimate works for $n=1$ and if we take $A= [0:2]^n$, then $|2A| = [0:4]^n$ and the bound fits. For $A \subset [0:d]^n$, I expect $|2A| \geq |A|^{\log_{d+1}{(2d+1)}}$. Are there any known results for such settings?