Waring-Hilbert problem on Cantor sets

Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set $C$. It is shown that, for each $m\ge 3$, every real number in the unit interval $0,1]$ is the sum $x_1^m+x_2^m+?+x_n^m$ with each $x_j$ in $C$ and some $n\le 6^m$. Furthermore, every real number $x$ in the interval $0,8]$ can be written as $x=x_1^3+x_2^3+?+x_8^3$, the sum of eight cubic powers with each $x_j$ in $C$. Another Cantor set $C\times C$ is also considered. More specifically, when $C\times C$ is embedded into the complex plane $?$, the Waring–Hilbert problem on $C\times C$ has a positive answer for powers less than or equal to 4.