This paper is part of the project of extending Davenport and Schmidt's result [DS] from 1970 on possible improvements of Dirichlet's Theorem. They proved that for Lebesgue almost every element of Mm,n≅R mn , Dirichlet's Theorem cannot be sigma-improved for any sigma 1. We are going to consider certain measures of "local maximal dimension" on [0, 1]mn. We want to prove for such a measure mu Dirichlet's Theorem cannot be sigma-improved for any sigma 1.;We are going to put these measures on the unstable submanifold of X=SLm+n,Z\ SLm+n,R under the right action of e-tmIn 00 etnIm with t > 0. Then we use entropy theory to prove that the orbit of such a measure is equidistributed with respect to the probability Haar measure on X. There is a difficulty in this approach, the limit of probability measures may not be a probability measure, that is, there might be loss of mass. This question is considered by Einsiedler and Kadyrov under weaker assumptions and they have some results for special cases. Friendly measures on Rn and dynamical systems with m = 1 are studied in [KLW]. Theorem 3.3 of that paper implies no loss of mass on average for a probability measure that is friendly.
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