On multi-dimensional continued fractions

In a recent breakthrough, a particular multi-dimensional analogue of the two-dimensional continued fraction algorithm has been established by Lettington and his collaborators, which incorporates and extends the normal continued fraction construction of convergents to a point \((x_{1},..., x_{m})\in\Bbb R^m\).

In the standard two-dimensional case, the convergents \(p_n/q_n\) to some real number \(\alpha\), follow a recurrence relation which can be written in matrix form such that:

\[\left(p_{r+1} \enspace p_{r} \atop q_{r+1} \enspace q_{r}\right )=\left (p_r \enspace p_{r-1}  \atop q_r \enspace q_{r-1} \right ) \left ({a_{r+1} \atop 1} {1 \atop 0} \right ) =\left (a_{r+1}p_r+p_{r-1} \enspace p_r \atop a_{r+1}q_r + q_{r-1}  \enspace q_r \right ).\]

Here the \(a_1\), \(a_2,\ldots\) are positive integers to be determined, and \(a_0=[\alpha]\), where by Dirichlet's theorem, consecutive convergents satisfy

\[
\left |\frac{p_n}{q_n}-\frac{p_{n+1}}{q_{n+1}}\right|= \frac{1}{q_n q_{n+1}},
\]
which guarantees an accuracy of
\[
\left|\frac{p_n}{q_n}-\alpha\right |\leq \frac{1}{q_n q_{n+1}}< \frac{1}{q_n^2}.
\]

With the context briefly outlined, we can now state the main aim of this PhD project, which is to establish analytic bounds for the accuracy of the convergents for the multi-dimensional continued fraction algorithm, alongside exploring geometric properties of lattice point constructions for these convergents.

We are interested in pursuing this project and welcome applications if you are self-funded or have funding from other sources, including government sponsorships or your employer.

Please contact the supervisor when you want to pursue this project, citing the project title in your email, or find out more about our PhD programme in Mathematics.

Supervisors

Photograph of Dr Matthew Lettington

Dr Matthew Lettington

Lecturer

Email:
lettingtonmc@cardiff.ac.uk
Telephone:
+44 (0)29 2087 5670

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For programme structure, entry requirements and how to apply, visit the Mathematics programme.

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