On arithmetic properties of Cantor sets

Abstract

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 2m. Moreover, we generalize this result to the middle-\({1 \over \alpha }\) Cantor set for \(1 < \alpha < 2 + \sqrt 5 \) 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 2m+8m-th powers of elements in W. At last, some similar results on p-adic Cantor sets are also obtained.