In this paper,we study three types of Cantor sets.For any integer m≥4,we show that every real number in[0,k]is the sum of at most k m-th powers of elements in the Cantor ternary set C for some positive integer k,and ...In this paper,we study three types of Cantor sets.For any integer m≥4,we show that every real number in[0,k]is the sum of at most k m-th powers of elements in the Cantor ternary set C for some positive integer k,and the smallest such k is 2~m.Moreover,we generalize this result to the middle-1/αCantor set for1<α<2+√5 and m sufficiently large.For the naturally embedded image W of the Cantor dust C×C into the complex plane C,we prove that for any integer m≥3,every element in the closed unit disk in C can be written as the sum of at most 2^(m+8)m-th powers of elements in W.At last,some similar results on p-adic Cantor sets are also obtained.展开更多
文摘In this paper,we study three types of Cantor sets.For any integer m≥4,we show that every real number in[0,k]is the sum of at most k m-th powers of elements in the Cantor ternary set C for some positive integer k,and the smallest such k is 2~m.Moreover,we generalize this result to the middle-1/αCantor set for1<α<2+√5 and m sufficiently large.For the naturally embedded image W of the Cantor dust C×C into the complex plane C,we prove that for any integer m≥3,every element in the closed unit disk in C can be written as the sum of at most 2^(m+8)m-th powers of elements in W.At last,some similar results on p-adic Cantor sets are also obtained.
