$n^2 α$小数部分间距的分布

缺项序列小数部分间距的分布

The distribution of spacings between fractional parts of lacunary sequences

给定一个递增的整数序列a（n），已知（由于Weyl），对于几乎所有实数t，扩张序列t*a（n）的小数部分在单位区间内均匀分布。最近已经做出了一些努力来理解这样的序列的元素之间的间距的分布（归一化为具有单位平均值），特别是对于多项式序列。在本文中，我们考虑缺项序列，例如2的幂的序列（即a（n）= 2^n）。我们证明了，对于几乎每一个t，序列t* 2 ^n的所有局部相关函数都是随机序列（泊松模型）的相关函数。除此之外，这意味着最近的邻居之间的间距，归一化为具有单位均值，具有指数分布。

The distribution of spacings between the fractional parts of $\boldsymbol{n^dα}$

We study the distribution of spacings between the fractional parts of $n^dα$. For $α$ of high enough Diophantine type we prove a necessary and sufficient condition for $n^dα\mod 1, 1\leq n\leq N,$ to be Poissonian as $N\to \infty$ along a suitable subsequence.

多维量子谐振子的能级间距统计：代数情形

Level spacing statistics for the multi-dimensional quantum harmonic oscillator: algebraic case

针对频率比生成代数数域的多维量子谐振子，本文利用代数数论证明重标度能级间距、相邻间距比及有限间距词的分布均随 log E 渐近准周期。

Nearest neighbor spacing distributions for zeros of the real or imaginary part of the Riemann xi-function on vertical lines

We show that the density functions of nearest neighbor spacing distributions for zeros of the real or imaginary part of the Riemann xi-function on vertical lines are described by the M-function which is appeared in value distributions of the logarithmic derivative of the Riemann zeta-function on vertical lines.

Nearest Neighbor Distances on a Circle: Multidimensional Case

We study the distances, called spacings, between pairs of neighboring energy levels for the quantum harmonic oscillator. Specifically, we consider all energy levels falling between E and E+1, and study how the spacings between these levels change for various choices of E, particularly when E goes to infinity. Primarily, we study the case in which the spring constant is a badly approximable vector. We first give the proof by Boshernitzan-Dyson that the number of distinct spacings has a uniform bound independent of E. Then, if the spring constant has components forming a basis of an algebraic nu…

Gaps between logs

We calculate the limiting gap distribution for the fractional parts of log n, where n runs through all positive integers. By rescaling the sequence, the proof quickly reduces to an argument used by Barra and Gaspard in the context of level spacing statistics for quantum graphs. The key ingredient is Weyl equidistribution of irrational translations on multi-dimensional tori. Our results extend to logarithms with arbitrary base; we deduce explicit formulas when the base is transcendental or the r:th root of an integer. If the base is close to one, the gap distribution is close to the exponential…