Diophantine Inequalities

This is the first book in a new series of London Mathematical Society Monographs. In a sequence n 2 with n=1,...,N, elements can be found that are close to zero modulo one. Starting with the work of I.M. Vinogradov and H. Heilbronn, the author develops the theme of nonlinear Diophantine approximation in a number of different directions. For example, one can consider simultaneous approximation to integers by values of a set of quadratic forms; or a discrete analogue (small solutions of a system of homogeneous congruences). This monograph gives an account of the most important and interesting results of the last ten years in this area at a leisurely pace that should be accessible to postgraduate students but also of value to research workers in analytic number theory.