Dimensions of Intersections, Exercise 3 Answers

Denote the Cantor middle halves set by H and the Cantor middle thirds set by T.

Recall the intersection formula: for almost all placements of S and L in E-dimensional space,

dim(H ∩ T) = dim(H) + dim(T) - E

For E = 1 this becomes

dim(H ∩ T) = Log(2)/Log(4) + Log(2)/Log(3) - 1 ≈ 0.131

For E = 2 this becomes

dim(H ∩ T) = Log(2)/Log(4) + Log(2)/Log(3) - 2 ≈ -0.869

Note for E = 1 if Cantor sets A and B have dimension a < 0.5 and b < 0.5, most placements of the Cantor sets have intersection with negative dimension, hence empty intersection.

dim(A ∩ B) = a + b - 1 < 0.

That is, the gaps in the Cantor sets are so substantial that for almost all placements, the points of one fall into the gaps of the other.

Return to Dimension Algebra Exercises.