math.NT

# Title: On arithmetic properties of Cantor sets

Authors: Lu Cui, Minghui Ma
Abstract: Three types of Cantor sets are studied.For any integer $m\ge 4$, 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 middle-$\frac 1\alpha$ Cantor set for $1<\alpha<2+\sqrt{5}$ and $m$ sufficiently large.For the naturally embedded image $W$ of the Cantor dust $C\times C$ into the complex plane $\mathbb{C}$, we prove that for any integer $m\ge 3$, every element in the closed unit disk in $\mathbb C$ 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.
 Subjects: Number Theory (math.NT) MSC classes: 28A80, 11P05 DOI: 10.1007/s11425-000-0000-0 Cite as: arXiv:2111.05489 [math.NT] (or arXiv:2111.05489v1 [math.NT] for this version)

## Submission history

From: Lu Cui [view email]
[v1] Wed, 10 Nov 2021 02:28:50 GMT (21kb)

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