arXiv 1610.060052016-10math.NT

关于丢番图逼近谱的拓扑

On the topology of Diophantine approximation Spectra

Damien Roy

备注：: 35 pages, 8 figures, minor corrections since version 1
发表信息：: Compositio Math. 153 (2017) 1512-1546
DOI：: 10.1112/S0010437X17007126

摘要：

固定一个整数$n\ge 2$。对于$\mathbb{R}^n$中的每个非零点$\mathbf{u}$，人们附加了几个称为丢番图逼近指数的数字。然而，正如欣钦首先观察到的那样，这些数字并不是相互独立的。这就产生了一个问题，即通过改变点$\mathbf{u}$来描述一个给定指数族所能取的所有可能值的集合。为了避免琐碎，限制为点$\mathbf{u}$，其坐标在$\mathbb{Q}$上线性无关。所得的值集称为这些指数的谱。我们证明，在适当的设置，任何这样的谱是一个紧凑的连通集。在$n=3$的情形下，我们进一步证明了它是一个半代数集，在分量最小下是闭的.对于$n=3$，我们也得到了一个家庭的六个指数最近推出的施密特和Summerer的频谱的描述。

英文摘要：

Fix an integer $n\ge 2$. To each non-zero point $\mathbf{u}$ in $\mathbb{R}^n$, one attaches several numbers called exponents of Diophantine approximation. However, as Khintchine first observed, these numbers are not independent of each other. This raises the problem of describing the set of all possible values that a given family of exponents can take by varying the point $\mathbf{u}$. To avoid trivialities, one restricts to points $\mathbf{u}$ whose coordinates are linearly independent over $\mathbb{Q}$. The resulting set of values is called the spectrum of these exponents. We show that, in an appropriate setting, any such spectrum is a compact connected set. In the case $n=3$, we prove moreover that it is a semi-algebraic set closed under component-wise minimum. For $n=3$, we also obtain a description of the spectrum of a family of six exponents recently introduced by Schmidt and Summerer.

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