Ergodic theory on $SL(n)$, Diophantine approximations and the lattice point counting in polyhedrons

We show that the lattice point counting for polyhedrons can be reduced to simultaneous diophantine approximations for linear forms. Next we interpret these diophantine problems in terms of certain flows on homogenious spaces $SL(n,R)/SL(n,Z)$. As a result we derive from ergodic theorems on semisimple groups that for any expanding polyhedron the lattice point counting has a logarithmically small error for almost all lattices. Applications of this result to algebraic number fields, theory of uniform distributions and spectral theory are also given.