On arithmetic sums of Cantor-type sequences of integers

We are looking for integer sets that resemble classical Cantor set and investigate the structure of their sum sets. Especially we investigate $FS(B)$ the subset sum of sequence type $B=\{\lfloor p^n\alpha \rfloor {\}}_{n=0}^{\mathrm{\infty }}$. When $p=2$, then we prove $FS(B)+FS(B)=\mathbb{N}$ by analogy with the Cantor set, and some structure theorem for $p>2$