LAUSR

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 x 1 m + x 2 m + ⋯ + x n m with each x j in C and some n ≤ 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 × 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.