高维随机游动中的可见格点及其偏差
Visible lattice points in higher dimensional random walks and biases among them
摘要：
对任意整数$k\geq 2$，$q\geq 1$和任意有限集$\mathcal{A}=\{{\boldsymbolα}_1,\cdots,{\boldsymbolα}_q\}$，其中${ \boldsymbolα_t}=(α_{t,1},\cdots,α_{t,k})~(1\leq t\leq q)$有$0<α_{t,1},\cdots,α_{t,k}<1$和$α_{t,1}+\cdots+α_{t,k}=1$，本文研究了格$\mathbb{Z}^k$上$\mathcal{A}$型随机游动中格点的可见性.我们证明了随机路径上的可见格点的比例几乎必然是$1/ζ(k)$，其中$ζ(s)$是Riemann zeta函数，我们还考虑了连续可见的格点的类型- $\mathcal{A}$随机游动，并给出相应的可见步骤的比例。此外，我们发现了一个新的现象，可见的步骤在上述两种情况下是不均匀分布。我们的证明依赖于概率论和解析数论的工具。
英文摘要：
For any integers $k\geq 2$, $q\geq 1$ and any finite set $\mathcal{A}=\{{\boldsymbolα}_1,\cdots,{\boldsymbolα}_q\}$, where ${ \boldsymbolα_t}=(α_{t,1},\cdots,α_{t,k})~(1\leq t\leq q)$ with $0<α_{t,1},\cdots,α_{t,k}<1$ and $α_{t,1}+\cdots+α_{t,k}=1$, this paper concerns the visibility of lattice points in the type-$\mathcal{A}$ random walk on the lattice $\mathbb{Z}^k$. We show that the proportion of visible lattice points on a random path of the walk is almost surely $1/ζ(k)$, where $ζ(s)$ is the Riemann zeta-function, and we also consider consecutive visibility of lattice points in the type-$\mathcal{A}$ random walk and give the proportion of the corresponding visible steps. Moreover, we find a new phenomenon that visible steps in both of the above cases are not evenly distributed. Our proof relies on tools from probability theory and analytic number theory.
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