Abstract
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≥3, every real number in the unit interval [0,1] is the sum x1m+x2m+⋯+xnm with each xj in C and some n≤6m. Furthermore, every real number x in the interval [0,8] can be written as x=x13+x23+⋯+x83, the sum of eight cubic powers with each xj in C. Another Cantor set C×C is also considered. More specifically, when C×C is embedded into the complex plane ℂ, the Waring–Hilbert problem on C×C has a positive answer for powers less than or equal to 4.
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