Some results in Diophantine approximation and some approaches that do not work

By Steffen Højris Pedersen
PhD Dissertations
October 2016
Abstract:

This thesis consists of three papers in Diophantine approximation, a subbranch of number theory. Preceding these papers is an introduction to various aspects of Diophantine approximation and formal Laurent series over \(\mathbb{F}_q\) and a summary of each of the three papers.

The introduction introduces the basic concepts on which the papers build. Among other it introduces metric Diophantine approximation, Mahler's approach on algebraic approximation, the Hausdorff measure, and properties of the formal Laurent series over \(\mathbb{F}_q\). The introduction ends with a discussion on Mahler's problem when considered in the formal Laurent series over \(\mathbb{F}_3\).

The first paper is on intrinsic Diophantine approximation in the Cantor set in the formal Laurent series over \(\mathbb{F}_3\). The summary contains a short motivation, the results of the paper and sketches of the proofs, mainly focusing on the ideas involved. The details of the proofs are in the paper.

The second paper is on higher dimensional Mahler approximation. The summary follows the same structure as in the case of the first paper.

The third paper is on twisted inhomogeneous Diophantine approximation in the formal Laurent series over \(\mathbb{F}_q\). The summary consists of two distinct parts. The first part is about a failed attempt of applying dynamical methods to obtain results and is not part of the paper. It explains the ideas of how the real case works and what goes wrong in the case of the formal Laurent series. The second part contains the results of the paper and sketches of the proofs.

Thesis advisor: Simon Kristensen
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