Sub plan on Hausdorff measure and fractals.

[Liu]

The last chapter of the book “real analysis” by Elias Stein and Rami
Shakarchi is on Housdorff measure. The fractional dimension is very
basic in fractional geometry. I also have heard about the space-
filling curves for a really long time. Also Besicovitch sets have
something to do with Kakeya problem, which I want to learn in the near
future.

[Liu]

I am very interested in learning more about Kakeya set. The last
chapter of the book introduces the regulatity property enjoyed by all
measurable subsets of R^d except when d=2, due to the existence of
Kakeya set. I want to read more materials in this subject later. Here
is a thought after reading the proof: I hope to think about another
proof of the fact that m(C+aC)=0 for almost all a\in R, where C is
cantor set, with the dissection 1/2 instead of 1/3.