论文标题
真实价值腔序列的配对相关函数的度量理论
The metric theory of the pair correlation function of real-valued lacunary sequences
论文作者
论文摘要
令$ \ {a（x）\} _ {x = 1}^{\ infty} $为正面的，真实的，可造的序列。本说明表明，扩张$αa（x）$的分数部分的对相关函数几乎是poissonian，几乎是\ mathbb {r} $的每一个$α\。通过使用谐波分析，我们的结果 - 不论选择实现的序列$ \ {a（x）\} _ {x = 1}^{\ infty} $ - 基本上可以简化以表明对二聚体不平等$ \ vert N_1（x_1）（x_1）（y_1（y_1）的解决方案的数量n_2（a（x_2）-a（y_2））\ vert <1 $ $ in Integer六个tuples $（n_1，n_2，n_2，x_1，x_1，x_2，y_1，y_1，y_2）$位于box $ [-n，n，n]^6 $的box $ [ - y_2，\ quad（n_1，n_2）\ neq（0,0），对于所有足够大的$ n $，对于某些固定$δ> 0 $，$$最多是$ n^{4-δ} $。
Let $\{ a(x) \}_{x=1}^{\infty}$ be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations $αa(x)$ is Poissonian for Lebesgue almost every $α\in \mathbb{R}$. By using harmonic analysis, our result - irrespective of the choice of the real-valued sequence $\{ a(x) \}_{x=1}^{\infty}$ - can essentially be reduced to showing that the number of solutions to the Diophantine inequality $$ \vert n_1 (a(x_1)-a(y_1))- n_2(a(x_2)-a(y_2)) \vert < 1 $$ in integer six-tuples $(n_1,n_2,x_1,x_2,y_1,y_2)$ located in the box $[-N,N]^6$ with the ``excluded diagonals'', that is $$x_1\neq y_1, \quad x_2 \neq y_2, \quad (n_1,n_2)\neq (0,0),$$ is at most $N^{4-δ}$ for some fixed $δ>0$, for all sufficiently large $N$.
