| In the thesis we prove some effective results in homogeneous dynamics and number theory. The thesis consists of two major parts. In the first part, we study the distribution of the Frobenius numbers. The Frobenius number F(a) of a lattice point a in R d with positive coprime coordinates, is the largest integer which can not be expressed as a non-negative integer linear combination of the coordinates of a. Marklof has recently proved the existence of the limit distribution of the Frobenius numbers, when a is taken to be random in an enlarging domain in R d. We will show that if the domain has piecewise smooth boundary, the error term for the convergence of the distribution function is at most a polynomial in the enlarging factor. The second part is devoted to the study of the reduction theory of ternary indefinite integral quadratic forms. We show that certain estimates in the classical reduction theory for integral quadratic forms can be improved, in an effective way, as long as the quadratic form is primitive and indefinite. The second part of the thesis is based on the joint work with my advisor Professor G. A. Margulis. |
