# The Sum of Two Cantor Sets

The Sum of Two Cantor Sets

The Cantor set is constructed iteratively. Starting with the closed unit interval , the open middle third , is taken out, leaving the two closed intervals and . Then the middle thirds of those two intervals are taken out, leaving four intervals of length , and so on. The Cantor set is the limit (or intersection) of all such sets.

C

[0,1]

1

3

2

3

0,

1

3

,1

2

3

1

9

Even though the Cantor set has measure zero and is nowhere dense, the set of sums where and are in is the whole interval , because the line always intersects the set .

x+y

x

y

C

[0,2]

x+y=a

C×C