论文标题
在较高的维泊松对相关性上
On higher dimensional Poissonian pair correlation
论文作者
论文摘要
在本文中，我们研究了较高维序列的对相关统计量。我们表明，对于任何$ d \ geq 2 $，严格增加序列$（a_n^{（1）}），\ ldots，（a_n^{（d））自然数的$具有公制的poissonian对相关，相对于sup-norm，如果它们的关节添加能量为$ o（n^3- {3- {3- {3- {3-}）$ $ $ $ $ $此外，在二维中，我们建立了相似的结果，相对于$ 2 $ norm。结果，得出的是，$（\ {nα\}，\ {n^2β\}）$和$（\ {nα\}，\ {[n \ log^an]β\ \}）$（$ a \ in [1,2] $ in [1,2] $ in [1,2] $几乎是poissonian colorelation for poissonian colelelation in poissonian colelelation in $（β） \ Mathbb {r}^2 $相对于SUP-NORM和$ 2 $ -NORM。这给出了Hofer和Kaltenböck[15]提出的问题的负面答案。该证明使用“广义” GCD-SUMS的估计值。
In this article we study the pair correlation statistic for higher dimensional sequences. We show that for any $d\geq 2$, strictly increasing sequences $(a_n^{(1)}),\ldots, (a_n^{(d)})$ of natural numbers have metric Poissonian pair correlation with respect to sup-norm if their joint additive energy is $O(N^{3-δ})$ for any $δ>0$. Further, in two dimension, we establish an analogous result with respect to $2$-norm. As a consequence, it follows that $(\{nα\}, \{n^2β\})$ and $(\{nα\}, \{[n\log^An]β\})$ ($A \in [1,2]$) have Poissonian pair correlation for almost all $(α,β)\in \mathbb{R}^2$ with respect to sup-norm and $2$-norm. This gives a negative answer to the question raised by Hofer and Kaltenböck [15]. The proof uses estimates for 'Generalized' GCD-sums.
