If C is the Cantor set, C+C contains an open set.

[WHOAguitarninja]

This is a statement my professor made in class some time ago (as a means to show that C contains a Hamel basis) that seemed fairly innocent, but it's bothered me for awhile. I did some searching online, and it seems that C+C=[0,2]. There it was again stated that this is fairly easy to show, but they neglected to give any insight as to how one might show it. Is there something simple I'm missing?

 

[xaos]

my rough guess is that any base 3 number (between 0 and 2) can be decomposed into a sum of two cantor numbers. if the digit is a zero, or a two, leave it alone. if its a 1, break it onto a sum of 2's that add up to 1.

e.g. if its 0.12, then 0.0222...+0.02