缺项序列小数部分间距的分布
The distribution of spacings between fractional parts of lacunary sequences
摘要：
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。
英文摘要：
Given an increasing sequence of integers a(n), it is known (due to Weyl) that for almost all reals t, the fractional parts of the dilated sequence t*a(n) are uniformly distributed in the unit interval. Some effort has been made recently to understand the distribution of the spacings between elements of such sequences (normalized to have unit mean), especially for sequences of polynomials. In this note, we consider lacunary sequences, such as the sequence of of powers of 2 (that is a(n)= 2^n). We show that for almost every t, all local correlation functions of the sequence t*2^n are those of a random sequence (the Poisson model). Among other things, this implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution.
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