论文标题
在扩张分析曲线的稀疏点的图像上
On the image in the torus of sparse points on dilating analytic curves
论文作者
论文摘要
众所周知，$ \ Mathbb {r}^{2}/\ mathbb {z}^{2} $ radius $ρ$的$在平面中的图像变为$ρ\ to \ infty $。我们考虑以下这种现象的稀疏版本。从半径$ \ left \ {ρ_{n} \ right \} _ {n = 1}^{\ infty} $开始，它在$ \ indbb {r}/\ mathbb {z}，$中，它在$ \ intbb { $ \ mathbb {r}^{2}/\ mathbb {z}^{2} $ unity的$ n $'th根部由Angle $ω$旋转，并以$ρ_{n} $的因子扩张。我们证明，如果$ρ_{n} $在$ n $中以多种方式界定，那么这些稀疏集合的图像将变得均衡，而且，如果$ρ_{n} $任意快速生长，那么我们显示了几乎所有$ω$的均衡性。有趣的是，我们发现，在任何角度，都有一系列半径生长到$ \ infty $的速度，然后等于等分分布巨大失败的任何多项式。在更大的一般性中，我们证明了这种类型的结果，用于在$ \ mathbb {r}^{d} $中的变化分析曲线扩张。证明的一个新组成部分是使用O最低结构理论来控制指数总和。
It is known that the image in $\mathbb{R}^{2}/\mathbb{Z}^{2}$ of a circle of radius $ρ$ in the plane becomes equidistributed as $ρ\to\infty$. We consider the following sparse version of this phenomenon. Starting from a sequence of radii $\left\{ ρ_{n}\right\} _{n=1}^{\infty}$ which diverges to $\infty$ and an angle $ω\in\mathbb{R}/\mathbb{Z},$ we consider the projection to $\mathbb{R}^{2}/\mathbb{Z}^{2}$ of the $n$'th roots of unity rotated by angle $ω$ and dilated by a factor of $ρ_{n}$. We prove that if $ρ_{n}$ is bounded polynomially in $n$, then the image of these sparse collections becomes equidistributed, and moreover, if $ρ_{n}$ grows arbitrarily fast, then we show that equidistribution holds for almost all $ω$. Interestingly, we found that for any angle there is a sequence of radii growing to $\infty$ faster then any polynomial for which equidistribution fails dramatically. In greater generality, we prove this type of results for dilations of varying analytic curves in $\mathbb{R}^{d}$. A novel component of the proof is the use of the theory of o-minimal structures to control exponential sums.
