Unconventional height functions in simultaneous Diophantine approximation

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• 作者：Lior Fishman ; David Simmons
• 关键词：Diophantine approximation ; Height functions ; Dirichlet’s theorem ; Continued fractions ; Hardy L ; functions
• 刊名：Monatshefte für Mathematik
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：182
• 期：3
• 页码：577-618
• 全文大小：711KB
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081
• 卷排序：182

文摘

Simultaneous Diophantine approximation is concerned with the approximation of a point \(\mathbf x\in \mathbb R^d\) by points \(\mathbf r\in \mathbb Q^d\), with a view towards jointly minimizing the quantities \(\Vert \mathbf x - \mathbf r\Vert \) and \(H(\mathbf r)\). Here \(H(\mathbf r)\) is the so-called “standard height” of the rational point \(\mathbf r\). In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examples of nonstandard height functions, computing their exponents of irrationality as well as giving more precise results. A list of open questions is also given.