论文标题
实价腔序列的中间尺度统计
Intermediate-scale statistics for real-valued lacunary sequences
论文作者
论文摘要
我们研究了序列$（αA_n）_ {n = 1}^{\ infty} $的中间尺度统计，其中$（a_n）_ {n = 1}^{\ infty} $是一个积极的，现实价值的差异序列，和$α\ in \ in \ in \ n rbbbbbbbbbbbb。特别是，我们考虑$ s_ {n}（l，α）$的元素的数量，在长度为$ l/n $的随机间隔中，其中$ l = o \ weft（n^{1-ε} \ right）$，并表明其差异（数字差异）是$ l $具有$ l $具有高概率w.r.r.t的$ l $的。 $α$，与统一的统计数据一致单位间隔中的随机点。此外，我们表明，当$ l = o \ left（n^{1/2-ε} \ right）$时，同一渐近学几乎肯定在$α\ in \ mathbb {r} $中。对于慢慢生长的$ l $，我们进一步证明了$ s_ {n}（l，α）$的中心限制定理，该定理几乎适用于\ mathbb {r} $中的几乎所有$α\。
We study intermediate-scale statistics for the fractional parts of the sequence $(αa_n)_{n=1}^{\infty}$, where $(a_n)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $α\in\mathbb{R}$. In particular, we consider the number of elements $S_{N}(L,α)$ in a random interval of length $L/N$, where $L=O\left(N^{1-ε}\right)$, and show that its variance (the number variance) is asymptotic to $L$ with high probability w.r.t. $α$, which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotics holds almost surely in $α\in\mathbb{R}$ when $L=O\left(N^{1/2-ε}\right)$. For slowly growing $L$, we further prove a central limit theorem for $S_{N}(L,α)$ which holds for almost all $α\in\mathbb{R}$.
