An Extension of Quantitative Nondivergence and Applications to Diophantine Exponents

This paper continues the theme started in 1971 by G.A. Margulis [22] when he showed that trajectories of one-parameter unipotent flows on SL(k,R)/ SL(k,Z) are never divergent. This was earlier conjectured by I. Piatetski-Shapiro, and was used by Margulis for the proof of the Arithmeticity Theorem for non-uniform lattices. A decade later, S. G. Dani [7, 8, 10] modified the method of Margulis, showing that any unipotent orbit returns to big compact subsets with high frequency. The latter statement was used in Dani’s proof [8] of finiteness of locally finite unipotentinvariant ergodic measures, and in M. Ratner’s proof [26, 27] of Raghunathan’s topological conjecture. The next development came in 1998, when a very general explicit estimate for the above frequency in terms of the size of compact sets was given in the paper of Margulis and the first named author [18]. In fact it was done in a bigger generality, namely for a large class of maps from R into SL(k,R), which made it possible to derive important applications to metric Diophantine approximation on manifolds. To state some of the results from that paper, recall that the space