MathDB
CIIM 2010 Problem 3

Source:

June 9, 2016
CIIMCIIM 2010undergraduate

Problem Statement

A set X⊂RX\subset \mathbb{R} has dimension zero if, for any ϵ>0\epsilon > 0 there exists a positive integer kk and intervals I1,I2,...,IkI_1,I_2,...,I_k such that X⊂I1∪I2∪⋯∪IkX \subset I_1 \cup I_2 \cup \cdots \cup I_k with ∑j=1k∣Ij∣ϵ<ϵ\sum_{j=1}^k |I_j|^{\epsilon} < \epsilon.
Prove that there exist sets X,Y⊂[0,1]X,Y \subset [0,1] both of dimension zero, such that X+Y=[0,2].X+Y = [0,2].
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