On complex analogues of real problems

By Rune Esdahl-Schou
PhD Dissertations
December 2013

This thesis will be a mix of different problems in number theory. As such it is split into two natural parts. The rst part focuses on normal numbers and construction of numbers that are normal to a given complex base. It is written in the style of a thorough and introductory paper on that subject. Certain classical theorems are stated without proof but with a reference instead, though usually a proof is given. This part of the thesis represents the pinnacle of the authors work during the first two years of his PhD study. The work presented is greatly inspired by the work of Madritsch, Thuswaldner and Tichy in [Madritsch et al., 2008] and [Madritsch, 2008] and contains a generalisation of the main theorem in [Madritsch, 2008].

The second part of the thesis focuses on Diophantine approximation, mainly on a famous conjecture by Schmidt from the 1980s. This conjecture was solved by Badziahin, Pollington and Velani, and inspired by this An gave a different proof which provides a stronger result. The conjecture is concerned with intersections of certain sets in the plane and are as such a real problem. We will consider a slightly different setup where the real plane is replaced by the complex plane. Using geometrical interpretations we construct sets with properties similar to the sets considered in the real case. We then formulate a conjecture which can be interpreted as a complex version of Schmidt's original conjecture. Finally we construct a variant of Schmidt's game, to show a partial result leading us to believe that the complex version of Schmidt's conjecture might some day be answered in the affirmative, just as the real one has.

Thesis advisor: Simon Kristensen
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