Dimension doubling theorem

In probability theory, the dimension doubling theorems are two results about the Hausdorff dimension of an image of a Brownian motion. In their core both statements say, that the dimension of a set under a Brownian motion doubles almost surely.
The first result is due to Henry P. McKean jr and hence called McKean's theorem. The second theorem is a refinement of McKean's result and called Kaufman's theorem since it was proven by Robert Kaufman.

Dimension doubling theorems

Let be a probability space. For a -dimensional Brownian motion and a set we define the image of under, i.e.

McKean's theorem

Let be a Brownian motion in dimension. Let, then
-almost surely.

Kaufman's theorem

Let be a Brownian motion in dimension. Then -almost surely, for any set, we have

Difference of the theorems

The difference of the theorems is the following: in McKean's result the -null sets, where the statement is not true, depends on the choice of. Kaufman's result on the other hand is true for all choices of simultaneously. This means Kaufman's theorem can also be applied to random sets.