Abstract
1. Introduction. In this paper we consider several related problems of the theory of Diophantine approximation. The following notation will be used. We denote by #S the number of elements in a finite set S. The Lebesgue measure of a measurable set S R is denoted by |S|. The set S R has full measure means that |R S| = 0. Throughout the paper, denotes a monotonic sequence of positive numbers. We denote by Pn the set of integral polynomials of degree n. The set of real algebraic numbers of degree n is denoted byAn. Given a polynomial P , H(P ) denotes the height of P . Given an algebraic number , H( ) denotes the height of . We use the Vinogradov symbol , which means “ up to a constant multiplier”. We begin with a short review. In 1924 Khinchin proved a remarkable result on the approximation of real numbers by rationals [9]. According to his theorem, for almost all x2R the inequality
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