论文标题
在$σ$ -pair的相关密度上，二次序列modulo一个
On the $σ$-Pair Correlation Density of Quadratic Sequences Modulo One
论文作者
论文摘要
在此注释中，我们研究$σ$ -PAIR相关密度\ begin {qore*} r_2^σ（[[a，b]，\ {θ_n\} _ n，n，n，n）= \ frac {1} {1} {n^} {n^{2- = {2- = {2-}}}}}}}}}}}}}} \，θ_{j} - θ_{k} \ in \ big [\ frac {a} {n^σ}，\ frac {b} {n^σ} \ big]+ \ big big bb z \ mathbb z \ big \} e e e e equence $ nisece $ nise $ \ n umite $ \ n umit Modulo One以$ 0 \ leqσ<2 $。 情况$σ= 1 $通常称为配对相关密度，序列$ \ {n^2α\} _ n $由于与贝里的猜想以及对通用性完全集成系统的能量水平的联系而引起了特别的兴趣。 我们证明，如果$α$是每$ε> 0 $的$ 3-ε$的二聚体，那么对于任何$ 0 \ leqleqσ<1 $ \ begin {align*} \ mathrm r_2^σ（[a，b] \ end {align*}在这种情况下，我们说序列表现出$σ$ -PAIR相关性。 除此之外，我们还表明，对于任何$ 0 \ leqσ<\ frac {1} {4}（9 - \ sqrt {17}）= 1.21922 ... $有一组完整的lebesgue量度，使得序列$ \ \ \ \ \ {αn^2 \} _ n $ corrip $ -ppiair。
In this note we study the $σ$-pair correlation density \begin{equation*}R_2^σ([a,b], \{ θ_n \}_n, N)= \frac{1}{N^{2-σ}} \# \big \{ 1 \leq j \neq k \leq N \, \big| \, θ_{j} - θ_{k} \in \big [ \frac{a}{N^σ},\frac{b}{N^σ} \big ]+ \mathbb Z \big \} \end{equation*} of a sequence $\{ θ_n\}_n$ that is equidistributed modulo one for $0 \leq σ<2$. The case $σ=1$ is commonly referred to as the pair correlation density and the sequence $\{ n^2 α\}_n$ has been of special interest due to its connection to a conjecture of Berry and Tabor on the energy levels of generic completely integrable systems. We prove that if $α$ is Diophantine of type $3-ε$ for every $ε>0$, then for any $0 \leq σ<1$ \begin{align*} \mathrm R_2^σ([a,b], \{ αn^2 \}_n, N) \to b-a, \text{ as } N \to \infty. \end{align*} In this case, we say that the sequence exhibits $σ$-pair correlation. In addition to this, we show that for any $0 \leq σ< \frac{1}{4}(9 -\sqrt{17})=1.21922...$ there is a set of full Lebesgue measure such that the sequence $\{ αn^2 \}_n$ exhibits $σ$-pair correlation.
