Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\}} = 2s.$$ It is known that this implies uniform distribution of the sequence $(x_n)$. Hinrichs, Kaltenbock, Larcher, Stockinger \& Ullrich extended this result to higher dimensions and showed that sequences $(x_n)$ in $[0,1]^d$ that satisfy, for all $s>0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: \|x_m - x_n\|_{\infty} \leq \frac{s}{N} \right\}} = (2s)^d.$$ are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence $(x_n)$ in $\mathbb{T}^d$ satisfies, for all $s > 0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: \|x_m - x_n\|_{2} \leq \frac{s}{N} \right\}} = \omega_d s^d$$ where $\omega_d$ is the volume of the unit ball, then $(x_n)$ is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.

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更高维度的泊松对相关

令 $(x_n)_{n=1}^{\infty}$ 是圆环 $\mathbb{T}$ 上的一个序列（标准化为长度 1）。如果对于所有 $s>0$，$$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\}} = 2s.$$ 众所周知，这意味着 $(x_n)$ 序列的均匀分布。Hinrichs, Kaltenbock, Larcher, Stockinger \& Ullrich 将这个结果扩展到更高的维度，并表明 $[0,1]^d$ 中的序列 $(x_n)$ 满足，对于所有 $s>0$，$$ \lim_ {N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: \|x_m - x_n\|_{\infty} \leq \frac{ s}{N} \right\}} = (2s)^d.$$ 也是均匀分布的。我们通过欧几里德范数证明了同样的扩展结果：如果 $\mathbb{T}^d$ 中的序列 $(x_n)$ 满足，对于所有 $s > 0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: \|x_m - x_n \|_{2} \leq \frac{s}{N} \right\}} = \omega_d s^d$$ 其中 $\omega_d$ 是单位球的体积，那么 $(x_n)$ 是一致的分散式。我们的方法表明泊松对相关意味着一个指数和估计，类似于并暗示 Weyl 标准。