I. The Four Gap Theorem and II. Differences of random Cantor sets

Abstract

This thesis consists of two parts, which are separate with respect to content. The first part considers a subject in the field of number theory, while in the second part a subject from probability theory is studied. The first part of this thesis deals with a variation on the Three Gap Theorem. The Three Gap Theorem states that the fractional parts of the first n multiples of an irrational number divide the interval [0,1] in subintervals of at most three different lengths. Instead of the fractional parts of these multiples, we considered the distances to the nearest integers, so the main question in this part of the thesis is: What can we say about the distribution of the distances of multiples of an irrational number to the nearest integer? We found a result similar to the Three Gap Theorem: these distances divide the interval [0,1/2] in subintervals of at most four different lengths. We give a proof of this result and also some additional properties are derived. The second part of this thesis is devoted to differences of random Cantor sets. The main question is the following: Under which conditions does the algebraic difference between two random Cantor sets contain an interval? There exist already some results in this direction. Dekking and Kuijvenhoven found some conditions under which the algebraic difference of two random Cantor sets contains an interval almost surely. In particular, they formulated the joint survival condition which they need to prove their main result. In the first chapter of Part II we try to find weaker conditions under which the result of Dekking and Kuijvenhoven can be proved. This results in the triangle growth condition and the max-min growth condition, of which the latter is the most promising condition. In the second chapter we consider a canonical class of random Cantor sets: correlated fractal percolation. Due to the elegant properties of this class, it is justified to pay special attention to it, but it also serves as a test case for the max-min growth condition. After derivation of some general results for correlated fractal percolation, we study some cases of flimsy correlated fractal percolation.