Geometric invariants of lattices and points close to a line, and their asymptotics
Barak Weiss (Tel-Aviv University)
Thursday, October 31, 2019, 11:10 – 12:00, -101
Given a lattice $\Lambda$ and a (perhaps long) vector $v \in \Lambda$, we consider two geometric quantities: - the projection $\Delta$ of $\Lambda$ along the line through $v$ - the “lift functional” which encodes how one can recover $\Lambda$ from the projection $\Delta$ Fixing $\Lambda$ and taking some infinite sequences of vectors $v_n$, we identify the asymptotic distribution of these two quantities. For example, for a.e. line $L$, if $v_n$ is the sequence of $\epsilon$-approximants to $L$ then the sequence $\Delta(v_n)$ equidistributes according to Haar measure, and if $v'_n$ is the sequence of best approximants to $L$ then there is another measure which $\Delta(v'_n)$ equidistributes according to. The basic tool is a cross section for a diagonal flow on the space of lattices, and after some analysis of this cross section, the results follow from the Birkhoff pointwise ergodic theorem.
Joint work with Uri Shapira.