A note on Cantor sets
type: Item
instance of: scholarly article
title: A note on Cantor sets
zbMATH Open document ID: 0943.26003
author: Eduardo J. Dubuc
published in: Real Analysis Exchange
publication date: 2000-05-21T00:00:00Z
review text: One of the puzzling properties of the classical Cantor set \(C\subset [0,1]\) is the fact that \(C+C=[0,2]\). This paper generalizes this property for results of more general constructions of Cantor sets, when given two real numbers \(0<a<b\), \(a+b<1\), and when each step consists of preserving the closed left \(100a\) percent and then the closed right \(100b\) percent of the interval(s) left in the preceding step (that is, \(a=1/3\) and \(b=1/2\) in the classical case); as usual, the starting interval is [0,1]. The author shows that the original geometric idea of the proof of \(C+C=[0,2]\) due to \textit{Steinhaus} fails if the deleted intervals are not equidistant from the end points, he modifies this argument and shows that \(C+\cdots +C=[0,n]\) (\(n\) summands on the left hand side) provided \((1-(a+b))(ab)^{-1}\leq n-1\).
Mathematics Subject Classification ID: 26A03
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 28A05
zbMATH DE Number: 1449460
zbMATH Keywords: Cantor ternary set
reviewed by: Miroslav Krbec
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1449460