Self-similar sets, projections and arithmetic sums

In this dissertation we consider problems concerning self-similar sets in the line and plane. We study the orthogonal projections of self-similar planar sets whose rotation angles are dense and prove that the gamma-Hausdorff measures of these projections are zero where gamma is the similarity dimension. We then consider arithmetic sums of the form Clambda + Cgamma where Clambda and Cgamma are self-similar sets in R . We give a necessary and sufficient condition for having H dlambda+ dgamma (Clambda + Cgamma) > 0 where dlambda, dgamma are the similarity dimensions of C lambda and Cgamma, respectively. Finally we prove a result about the local structure of the Mandelbrot set M associated to the linear system {lcub}lambdaz, lambdaz + 1{rcub}.