Special divergent trajectories for a homogeneous flow

Abstract. Let L be a unimodular lattice in R^{d+1} and consider
the evolution L_t of this lattice under the action of diag(e^t,…,e^t,e^{-dt}). The kth successive minimum of a lattice L, denoted lambda_k(L),
is the smallest possible length for the longest vector in a linearly
independent subset of L of order k. When k=1, this is simply the
shortest nonzero vector in L. Clearly, lambda_1(L)<=…<=lambda_{d+1}(L). It’s also not hard to see that the product of the successive minima
is universally bounded above and below by some positive constants. Schmidt conjectured that for any k=1,…,d-1 there exist L such that
lambda_k(L_t) tends to zero and lambda_{k+2}(L_t) tends to infinity
as t tends to infinity (with lambda_{k+1}(L_t) oscillating between the
two). We construct examples satisfying Schmidt conjecture. This is
joint work with Barak Weiss. (Earlier examples have also been
constructed by Moshchevitin.)