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.
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Acknowledgements
Our deepest gratitude goes first and foremost to Professor Liming Ge for his kind guidance and support. The authors also thank Bo Qi, Dongsheng Wu, Boqing Xue, Professor Wei Yuan, Yifeng Ye and Hanbin Zhang for their helpful suggestions and discussions.
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Cui, L., Ma, M. On arithmetic properties of Cantor sets. Sci. China Math. 65, 2035–2060 (2022). https://doi.org/10.1007/s11425-021-1924-1
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DOI: https://doi.org/10.1007/s11425-021-1924-1