Cutting Sequences and the p-adic Littlewood Conjecture

Abstract

The main aim of this thesis is to use the geometric setting of cutting sequences to better understand the behaviour of continued fractions under integer multiplication. We will use cutting sequences to construct an algorithm that multiplies continued fractions by an integer . The theoretical aspects of this algorithm allow us to explore the interesting properties of continued fractions under integer multiplication. In particular, we show that an eventually recurrent continued fractions remain eventually recurrent when multiplied by a rational number. Finally, and most importantly, we provide a reformulation the -adic Littlewood Conjecture in terms of a condition on the semi-convergents of a real number .