Generalized Continued Fractions and Differential Equations

A new theory of generalized continued fractions for vectors from real n-dimensional space is constructed. It is represented in two variants. The first variant is connected with some strictly ergodic transformations and the second variant is the generalization of first one and is connected with differential equations. This theory is applied to obtain solutions of some classical problems of analysis and number theory. In particular, by using this theory one finds strong estimates of upper bounds of Weyl sums, the remainder terms in the laws of distribution and joint distribution of fractional parts of polynomial values. The generalized continued fractions connected with differential equation allow characterizing of some class of algebraic numbers.