On Quasi-Energy-Spectra, Pair Correlations of Sequences and Additive Combinatorics
On Quasi-Energy-Spectra, Pair Correlations of Sequences and Additive Combinatorics
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关于准能量谱、序列对相关性和加性组合学
DOI：
10.1007/978-3-319-72456-0_1
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
G. Larcher
中科院分区：
文献类型：
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作者：
I. Aichinger;C. Aistleitner;G. Larcher
对序列配对相关统计的研究最初是由有关量子系统准能量谱的问题引发的。然而，该学科的发展远远超出了其数学物理学的根源，并且关于某些序列的配对相关性分布的许多具有挑战性的数论问题仍然悬而未决。我们对该主题进行简短介绍，回顾一些已知的结果和未解决的问题，并特别解释最近在环面上序列对相关性分布与加性组合学中的某些概念之间建立的联系。此外，我们稍微改进了 Jean Bourgain 最近在 Aistleitner 等人中给出的结果。 （Isr. J. Math.，即将出现。可在 https://arxiv.org/abs/1606.03591 获取）。
The investigation of the pair correlation statistics of sequences was initially motivated by questions concerning quasi-energy-spectra of quantum systems. However, the subject has been developed far beyond its roots in mathematical physics, and many challenging number-theoretic questions on the distribution of the pair correlations of certain sequences are still open. We give a short introduction into the subject, recall some known results and open problems, and in particular explain the recently established connection between the distribution of pair correlations of sequences on the torus and certain concepts from additive combinatorics. Furthermore, we slightly improve a result recently given by Jean Bourgain in Aistleitner et al. (Isr. J. Math., to appear. Available at https://arxiv.org/abs/1606.03591).