LOWER BOUNDS FOR THE NUMBER OF VECTORS WITH ALGEBRAIC COORDINATES NEAR SMOOTH SURFACES

Let z = f(x, y) be a surface in three-dimensional Euclidean space. Consider a neighborhood V of this surface, whose points satisfy the inequality vertical bar f (x, y) - z vertical bar < Q(-gamma), where 0 < gamma < 1 and Q is a sufficiently large positive integer. In the works of Huxley, Beresnevich, Velani, the distribution of rational points in V has been started. In this article, we study the distribution of points with real conjugate algebraic coordinates <(alpha)over right arrow> = alpha(1), alpha(2), alpha(3) in V. For some c(1) = c(1)(n), a lower bound is obtained in the form of c(2)Q(n+1-gamma) for the number of algebraic numbers of degree n >= 3 and of height at most c(3)Q.