This is my bachelor's thesis.
In my thesis, I study the Hausdorff measure and Hausdorff dimension in \( \mathbb{R}^n \). I begin by presenting the basic concepts and definitions and examining some of the most important properties of the Hausdorff measure, such as its relation with the Lebesgue measure. I later introduce the concept of Hausdorff dimension and work with similarities and self–similar sets in order to prove a result which grants the existence of such sets under specific conditions, along with a way to obtain their Hausdorff dimension.
During the second part of the project, I explore the relation between the Hausdorff dimension of a set in \( \mathbb{R}^n \) and the energy integrals of Borel measures which provide said set with positive, finite measure. For this part I also make use of some concepts from distribution theory. This allows me to find an identity involving energy integrals of measures and the Fourier transform, thereby establishing a relation between the Hausdorff dimension and the Fourier transform.
Lastly, I combine several results seen in previous chapters to study a geometric problem regarding Hausdorff dimension, and I prove an important theorem related to Falconer’s distance set conjecture, which states that if a Borel set A in \( \mathbb{R}^n \) has Hausdorff dimension strictly greater that \( n/2 \), then the set of distances between every pair of points in A has positive Lebesgue measure.