Ben Ward

The Lagrange spectrum in higher dimensions

In this talk I will give a brief survey on the Markoff-Lagrange spectrum and generalisations. In particular I will talk about the Lagrange spectrum for simultaneous Diophantine approximation in dimension $d\geq 2$. For any $d\in \N_{\geq 2}$ and arbitrary norm $\|\cdot\|$ on $\R^{d}$ define
\begin{equation*}
\Theta_{d}(\bx, \|\cdot\|):=\liminf_{q\to\infty}\left(q^{\tfrac{1}{d}}\min_{\bp\in\Z^{d}}\|q\bx-\bp\|\right)
\end{equation*}
We show there exists constants $C,\delta>0$, depending on $d$ and $\|\cdot\|$, such that for every $0<\varepsilon