Student Seminar Talk HAUSDORFF DIMENSION

We are familiar with the idea that a curve is a 1-dimensional object and a surface is 2-dimensional. What is the “dimension” of a set having infinite length but zero area? Similarly, the Cantor set has Lebesgue measure zero, but in some ways it is quite large (it is uncountable). How to quantify its size? In this talk we will answer these questions through the notions of Hausdorff measure and dimension. We will also describe methods for calculation of Hausdorff dimension and consider some examples. 1. Box-counting dimension We work in the d-dimensional Euclidean space R. We start with the following Definition 1.1. The ε-covering number of a bounded set E ⊂ R is N(E, ε) = min{N : ∃B1, . . . , BN balls of diameter ε such that E ⊂ N ⋃