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.

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 .