论文标题
线性团体动作及其产品的格拉斯纳属性
Glasner property for linear group actions and their products
论文作者
论文摘要
格拉斯纳（Glasner）的定理从1979年开始表明，如果$ y \ subset \ mathbb {t} = \ mathbb {r}/\ mathbb {z} $是无限的，那么每个$ε> 0 $都存在一个integer $ n $ n $ ny $ ny $是$ ny $ is $ε$ - $ - $ - $ - dense。在各种作品中，通过表明对$ \ mathbb {t}^d $上的某些不可约线的线性半群操作也扩展了这一点，也满足了这样一个\ textit {glasner property}，每个无限套件（实际上，任意大的有限设置）将具有$ε$密集的图像，这些元素来自Act Act Act ot Acting semigroup的某些元素。我们通过证明具有Zariski连接的Zariski-closure的不可还原线性群体作用的定量Glasner定理来改善这些作品。这利用了最新的结果在圆环上的线性随机步行中。我们还提出了一个自然的问题，询问满足格拉斯纳特性的两种动作的笛卡尔产物是否还满足了无限子集的Glasner特性，而Glasner属性是在常见的垂直或水平线上不包含两个点的。我们通过为不可约束的线性动作以及此类结果的多项式版本提供了新的Glasner型定理，以对许多此类Glasner行为进行肯定地回答这个问题。
A theorem of Glasner from 1979 shows that if $Y \subset \mathbb{T} = \mathbb{R}/\mathbb{Z}$ is infinite then for each $ε> 0$ there exists an integer $n$ such that $nY$ is $ε$-dense. This has been extended in various works by showing that certain irreducible linear semigroup actions on $\mathbb{T}^d$ also satisfy such a \textit{Glasner property} where each infinite set (in fact, arbitrarily large finite set) will have an $ε$-dense image under some element from the acting semigroup. We improve these works by proving a quantitative Glasner theorem for irreducible linear group actions with Zariski-connected Zariski-closure. This makes use of recent results on linear random walks on the torus. We also pose a natural question that asks whether the cartesian product of two actions satisfying the Glasner property also satisfy a Glasner property for infinite subsets which contain no two points on a common vertical or horizontal line. We answer this question affirmatively for many such Glasner actions by providing a new Glasner-type theorem for linear actions that are not irreducible, as well as polynomial versions of such results.
