Pair correlation of fractional parts derived from rational valued sequences, II

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• 作者：Sneha Chaubey^a ; ^{chaubey2@illinois.edu" class="auth_mail" title="E-mail the corresponding author} ; Melinda Lanius^a ; ^{lanius2@illinois.edu" class="auth_mail" title="E-mail the corresponding author} ; Alexandru Zaharescu^b ; ^a ; ^{zaharesc@illinois.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11K99 ; 11B99
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：158
• 期：Complete
• 页码：151-164
• 全文大小：338 K

文摘

A rational valued vector sequence $\overrightarrow{x}$, for some fixed k   and e153407" title="Click to view the MathML source">r∈N$r\in \mathbb{N}$, is a map $\overrightarrow{x}:{\mathbb{N}}^k\to {\mathbb{Q}}^r$. In the present paper, we complement the results of [3] with a discussion on rational valued vector sequences. We investigate the pair correlation function for the fractional parts of sequences $\overrightarrow{t}\cdot \overrightarrow{x}$, where $\overrightarrow{x}$ is a rational valued vector sequence and 30e0308ab9625a6aad1e6f">$\overrightarrow{t}\in {\mathbb{R}}^r$. We offer a new class of sequences $\overrightarrow{x}$ whose pair correlation function behaves as that of random sequences for almost all real vectors $\overrightarrow{t}$, namely, injective sequences of the form for p₁$p_1$, p₂$p_2$, p₃$p_3$, and p₄$p_4$ primes and where $\overrightarrow{\mathfrak{b}}:{\mathbb{N}}^2\to \mathbb{N}$ satisfies a certain growth condition.