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 $x_1^m+x_2^m+\cdots +x_n^m$ 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 $x=x_1^3+x_2^3+\cdots +x_8^3$, 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 $ℂ$, the Waring–Hilbert problem on $C\times C$ has a positive answer for powers less than or equal to 4.

中文翻译：

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康托集上的 Waring-Hilbert 问题

探索 Cantor 集上 Waring-Hilbert 问题的类比。本文的重点是康托三元集$C$. 表明，对于每个$米\ge 3$, 单位区间内的每个实数 $[0,1]$ 是总和 $X_1^{米}+X_2^{米}+\cdots +X_n^{米}$ 与每个 $X_j$ 在 $C$ 还有一些 $n\le 6^{米}$. 此外，每个实数$X$ 在区间 $[0,8]$ 可以写成 $X=X_1^3+X_2^3+\cdots +X_8^3$, 八次方的总和 $X_j$ 在 $C$. 另一个康托尔集$C\times C$也考虑。更具体地说，当$C\times C$ 嵌入到复平面中 $ℂ$, Waring-Hilbert 问题 $C\times C$ 对小于或等于 4 的幂有肯定的回答。