On approximation of real numbers by real algebraic numbers
On approximation of real numbers by real algebraic numbers
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用实代数数逼近实数
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
V. Beresnevich
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文献类型：
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作者：
V. Beresnevich
1.引言。本文考虑丢番图逼近理论的几个相关问题。将使用以下表示法。我们用#S表示有限集S中元素的个数，用|S|表示可测集S R的勒贝格测度。集合S R有满测度表示|R S|=0。在整篇文章中，表示一个单调的正数序列。我们用Pn表示n次整多项式的集合。n次实代数集表示为Byan。给定一个多项式P，H(P)表示P的高度。给定一个代数数，H()表示的高度。我们使用Vinogradov符号，意思是“最多一个恒定的乘数”。我们从简短的回顾开始。1924年，金钦证明了有理数逼近实数的显著结果[9]。根据他的定理，对于几乎所有的x2r，
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