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Martin Widmer

Rate of escape for lattices and their duales
Jeudi, 9 Janvier, 2020 - 10:30
Résumé :

For weakly admissible lattices $\Lambda\in SL_n(\mathbb{Z})\backslash SL_n(\mathbb{R})$ (i.e., lattices that have no non-zero lattice point on the coordinate subspaces) the rate of escape'' under the action of the diagonal matrices $\Delta_n\subset SL_n(\mathbb{R})$ is controlled by the function $\nu_{\Lambda}(\rho)=\inf\{|x_1\cdots x_n|; {\bf x}\in \Lambda, 0<|{\bf x}|<\rho\}$.

We are interested in relations between $\nu_{\Lambda}(\rho)$ and $\nu_{\Lambda^\perp}(\rho)$, where $\Lambda^\perp$ denotes the dual lattice (w.r.t. the standard inner product on $\mathbb{R}^n$). It is easy to see that  $\nu_{\Lambda}(\rho)$ tends to zero if and only if  $\nu_{\Lambda^\perp}(\rho)$ does. Can this be made quantitive? When do we have  $\nu_{\Lambda}(\rho)=\nu_{\Lambda^\perp}(\rho)$?
(These questions naturally turn up when comparing different error estimates for lattice point counting.)
We answer these questions, and we show how they relate to Beresnevich's result on Davenport's problem regarding badly approximable points on submanifolds of $\mathbb{R}^n$. This is joint work with Niclas Technau.

Institution de l'orateur :
Department of Mathematics Royal Holloway University of London
Thème de recherche :
Théorie des nombres
Salle :
SALLE 6