Diophantine approximation and Dynamical systems

By Jonas Lindstrøm Jensen
PhD Dissertations
October 2012

In this dissertation we consider approximation of real numbers by rationals with certain restrictions on the denominators. First we consider numbers that are badly approximable from the left by fractions with denominator of the form $q^n$ for a fixed integer $q \geq 2$ and all $n \geq 0$. More precisely we let $c \in (0,1)$ and consider the $x \in [0,1)$ such that

$x - \frac{k}{q^n} \geq \frac{c}{q^n} $
for all $n \geq 0$ and $\frac{k}{q^n} \leq x$. The set of these $x$ is a nullset with respect to the Lebesgue measure, and we give a formula on how to calculate the Hausdorff dimension of this set. This formula is then generalized to the case where $q = \beta > 1$ is in a certain dense set of real numbers, namely the simple numbers.

Then, we let $p,q \geq 2$ be integers and consider approximation by fractions with denominators of the form $p^n q^m$ for all $n,m \geq 0$ and prove some versions of classical results in this settings.

Finally we show a result related to the famous Littlewood conjecture. We prove that there is a subset of the badly approximable numbers of full Hausdorff dimension such that a family of conjectures related to the Littlewood conjecture is simultaneously true on this set, namely the Littlewood conjecture, the mixed Littlewood conjecture and a hybrid of a conjecture by Cassel and Littlewoods conjecture.

Thesis advisor: Simon Kristensen
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