AI 中文总结
该研究分析高维球随机平移下的格点计数渐近行为，在临界尺度附近得到中心极限定理与对数正态极限，临界尺度以上得到计数体积比的收敛性及归一化误差的正态分布与方差渐近公式。
AI 中文摘要
设d维欧氏球的球心在模ℤᵈ意义下均匀分布，我们研究维度与半径趋于无穷时对应的整数格点数目。临界尺度为4π²R_d²/(d+2)=log d。在该尺度附近的对数窗口内，格点计数的对数除以体积满足中心极限定理；在与临界尺度有固定偏移的情形下，可得到真正的对数正态极限。在临界尺度以上，计数与体积之比依测度收敛于1，其归一化误差服从标准正态极限，且方差满足一个渐近公式。
英文摘要
Let the center of a $d$-dimensional Euclidean ball be uniformly distributed modulo $\mathbb Z^d$. We study the resulting number of integer lattice points as the dimension and radius tend to infinity. The critical scale is $4π^2R_d^2/(d+2)=\log d$. In a logarithmic window around this scale, the logarithm of the lattice-point count divided by the volume satisfies a central limit theorem. At a fixed offset from the critical scale this yields a genuine lognormal limit. Above the critical scale, the count-to-volume ratio converges to $1$ in measure; its normalized error has a standard normal limit, and its variance satisfies an asymptotic formula.
