Fishman Lior, Simmons David
1Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX 76203-5017 USA.
2Department of Mathematics, University of York, Heslington, York YO10 5DD UK.
Mon Hefte Math. 2017;182(3):577-618. doi: 10.1007/s00605-016-0983-0. Epub 2016 Oct 18.
Simultaneous Diophantine approximation is concerned with the approximation of a point by points , with a view towards jointly minimizing the quantities and . Here is the so-called "standard height" of the rational point . 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.
联立丢番图逼近关注的是用点(p_n)逼近一个点(p)，目的是共同最小化量(\vert p - p_n\vert)和(H(p_n))。这里(H(p_n))是有理点(p_n)的所谓“标准高度”。在本文中，作者提出问题：如果我们用一个不同的高度函数替换标准高度函数会有什么变化？事实证明，这种变化会导致与经典理论有显著差异，并且需要开发新方法。我们讨论三个非标准高度函数的例子，计算它们的无理指数并给出更精确的结果。还给出了一系列未解决的问题。
