Infinite-dimensional generalized continued fractions, sums of Legendre symbols and distribution of quadratic residues and non-residues

A new theory of generalized continued fractions for infinite-dimensional vectors with integer components is constructed. The results of this theory are applied to the classical problems on the estimates of sums of Legendre symbols and on the distribution of quadratic residues and non-residues modulo a prime number. The proofs are based on the study of ergodic properties of some infinite-dimensional transformations.