On arithmetic properties of Cantor sets

TL;DR

This paper investigates the additive properties of various Cantor sets, demonstrating how real and complex numbers can be expressed as sums of powers of elements within these fractal structures, with specific bounds and generalizations.

Contribution

It establishes new bounds for representing numbers as sums of powers in Cantor sets and extends these results to complex and p-adic Cantor sets, generalizing previous work.

Findings

Every number in [0,k] can be expressed as a sum of at most 2^m m-th powers in the Cantor ternary set.

Generalization to middle-1/α Cantor sets for certain α and large m.

Every point in the unit disk can be written as a sum of at most 2^{m+8} m-th powers in the embedded Cantor dust.

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.