Waring-Hilbert problem on Cantor sets

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.