论文标题
差距统计数据和几何发展的较高相关性模型一个
Gap statistics and higher correlations for geometric progressions modulo one
论文作者
论文摘要
科克斯马（Koksma）的等分定理1935年指出，对于勒布斯格（Lebesgue）而言，几乎每$α> 1 $，几何进程的分数$（α^{n}）_ {n \ geq1} $是等分的modulo one。在本文中，我们通过证明几乎每$α> 1 $，所有有限订单的相关性，以及$（α^{n}）_ {n \ geq1} $ mod 1具有Poissonian限制分布，从而解决两个命名的授权者的命名。虽然一种早期的方法采用了概率方法，但本文中我们在本文中的推理具有分析性质，并且基于振荡积分的估计。该方法足够强大，可以使我们能够将结果扩展到天然的量距序列。
Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $α>1$, the fractional parts of the geometric progression $(α^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every $α>1$, the correlations of all finite orders and hence the normalized gaps of $(α^{n})_{n\geq1}$ mod 1 have a Poissonian limit distribution, thereby resolving a conjecture of the two first named authors. While an earlier approach used probabilistic methods in the form of martingale approximation, our reasoning in the present paper is of an analytic nature and based upon the estimation of oscillatory integrals. This method is robust enough to allow us to extend our results to a natural class of sub-lacunary sequences.
