在同时的丢番图近似中，非常规高度函数。

IF 0.8 4区数学 Q2 MATHEMATICS

Monatshefte fur Mathematik Pub Date : 2017-01-01 Epub Date: 2016-10-18 DOI:10.1007/s00605-016-0983-0

Lior Fishman, David Simmons

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引用次数: 3

摘要

同时Diophantine近似关注的是点x∈R d对点R∈Q d的近似，目的是使量‖x - R‖和H (R)共同最小化。这里H (r)是有理点r的所谓“标准高度”。在本文中，作者提出了一个问题:如果我们用一个不同的高度函数代替标准的高度函数，会有什么变化?事实证明，这种变化导致了与经典理论的巨大差异，需要发展新的方法。我们讨论了三个非标准高度函数的例子，计算了它们的非理性指数，并给出了更精确的结果。还列出了一些悬而未决的问题。

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Unconventional height functions in simultaneous Diophantine approximation.

Simultaneous Diophantine approximation is concerned with the approximation of a point $x \in R^{d}$ by points $r \in Q^{d}$ , with a view towards jointly minimizing the quantities $‖ x - r ‖$ and $H (r)$ . Here $H (r)$ is the so-called "standard height" of the rational point $r$ . 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.

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来源期刊

Monatshefte fur Mathematik 数学-数学

CiteScore

1.60

自引率

11.10%

发文量

155

审稿时长

4-8 weeks

期刊介绍： The journal was founded in 1890 by G. v. Escherich and E. Weyr as "Monatshefte für Mathematik und Physik" and appeared with this title until 1944. Continued from 1948 on as "Monatshefte für Mathematik", its managing editors were L. Gegenbauer, F. Mertens, W. Wirtinger, H. Hahn, Ph. Furtwängler, J. Radon, K. Mayrhofer, N. Hofreiter, H. Reiter, K. Sigmund, J. Cigler. The journal is devoted to research in mathematics in its broadest sense. Over the years, it has attracted a remarkable cast of authors, ranging from G. Peano, and A. Tauber to P. Erdös and B. L. van der Waerden. The volumes of the Monatshefte contain historical achievements in analysis (L. Bieberbach, H. Hahn, E. Helly, R. Nevanlinna, J. Radon, F. Riesz, W. Wirtinger), topology (K. Menger, K. Kuratowski, L. Vietoris, K. Reidemeister), and number theory (F. Mertens, Ph. Furtwängler, E. Hlawka, E. Landau). It also published landmark contributions by physicists such as M. Planck and W. Heisenberg and by philosophers such as R. Carnap and F. Waismann. In particular, the journal played a seminal role in analyzing the foundations of mathematics (L. E. J. Brouwer, A. Tarski and K. Gödel). The journal publishes research papers of general interest in all areas of mathematics. Surveys of significant developments in the fields of pure and applied mathematics and mathematical physics may be occasionally included.