The attached link is Senior Integrative Project in Carleton College that my group and I did over last term: COMPSFINAL. The project was underwent over Fall 2014 and Winter 2015 under supervision of Allison Tanguay.

The main purpose of the paper is to introduce the idea of Lebesgue measure and -dimensional Hausdorff measure, fractals and their behaviors, non-fractal sets and related conjecture.

Lee’s Challenge #1:

Consider an arbitrary non-negative real number, namely . Can we always construct a set of which strict Hausdorff dimension is ?