  Three types of Cantor sets are studied.For any integer $m\ge 4$, we show that
every real number in $[0,k]$ is the sum of at most $k$ $m$-th powers of
elements in the Cantor ternary set $C$ for some positive integer $k$, and the
smallest such $k$ is $2^m$.Moreover, we generalize this result to middle-$\frac
1\alpha$ Cantor set for $1<\alpha<2+\sqrt{5}$ and $m$ sufficiently large.For
the naturally embedded image $W$ of the Cantor dust $C\times C$ into the
complex plane $\mathbb{C}$, we prove that for any integer $m\ge 3$, every
element in the closed unit disk in $\mathbb C$ can be written as the sum of at
most $2^{m+8}$ $m$-th powers of elements in $W$.At last, some similar results
on $p$-adic Cantor sets are also obtained.
