论文标题
限制晶格问题中的法律。 iv。 $ \ mathbb {z}^{d} $的特殊情况
Limit laws in the lattice problem. IV. The special case of $\mathbb{Z}^{d}$
论文作者
论文摘要
我们研究了晶格$ \ mathbb {z}^{d} $的点的误差，该点属于$ 0 $左右的扩张和翻译的超立方体，其轴与坐标的轴平行。 We show that if $t$, the factor of dilatation, is distributed according to the probability measure $\frac{1}{T} ρ(\frac{t}{T}) dt$ with $ρ$ being a probability density over $[0,1]$ the error, when normalized by $t^{d-1}$, converges in law when $T \rightarrow \infty$ in the case where翻译的形式为$ x =（x，\ cdots，x）$，如果$ x $之间的坐标在它们之间是独立的，则独立于$ t $，并且根据$ [ - \ frac {1} {1} {2} {2} {2} {2} {2} {2}，\ frac {1} {1} {2} {2}] $分配。在这两种情况下，我们都计算极限定律的特征功能。
We study the error of the number of points of the lattice $\mathbb{Z}^{d}$ that fall into a dilated and translated hypercube centred around $0$ and whose axis are parallel to the axis of coordinates. We show that if $t$, the factor of dilatation, is distributed according to the probability measure $\frac{1}{T} ρ(\frac{t}{T}) dt$ with $ρ$ being a probability density over $[0,1]$ the error, when normalized by $t^{d-1}$, converges in law when $T \rightarrow \infty$ in the case where the translation is of the form $X=(x,\cdots,x)$ and in the case where the coordinates of $X$ are independent between them, independent from $t$ and distributed according to the uniform law over $[-\frac{1}{2},\frac{1}{2}]$. In both cases, we compute the characteristic function of the limit law.
