论文标题
限制晶格问题中的法律。 iii。返回盒子的情况
Limit laws in the lattice problem. III. Return to the case of boxes
论文作者
论文摘要
我们研究了属于矩形的晶格$ l $点数的误差，该矩形为$ 0 $，其轴平行于坐标轴，由因子$ t $扩张，然后由vector $ x \ in \ mathbb {r}^{r}^{2} $翻译。 When we consider the second order moment of the error relatively to $X \in \mathbb{R}^{2}/L$, one shows that, when $t$ is random and becomes large and when the error is normalized by a quantity which behaves, in the admissible case, as $\sqrt{\log(t)}$, it converges in distribution to an explicit positive constant.在典型的晶格$ l $的情况下，我们表明该结果仍然存在，但标准化更为重要，左右$ \ log（t）$。我们还表明，当$ l = \ mathbb {z}^{2} $时，当$ t $归一化时，当$ t $ native时，当$ t $是随机的时，分配会收敛，并且我们会变大，并且我们计算了极限分布的矩。
We study the error of the number of points of a lattice $L$ that belong to a rectangle, centred at $0$, whose axes are parallel to the coordinate axes, dilated by a factor $t$ and then translated by a vector $X \in \mathbb{R}^{2}$. When we consider the second order moment of the error relatively to $X \in \mathbb{R}^{2}/L$, one shows that, when $t$ is random and becomes large and when the error is normalized by a quantity which behaves, in the admissible case, as $\sqrt{\log(t)}$, it converges in distribution to an explicit positive constant. In the case of a typical lattice $L$, we show that this result still holds but the normalisation is more important, around $\log(t)$. We also show that when $L=\mathbb{Z}^{2}$, the error, when normalized by $t$, converges in distribution when $t$ is random and becomes large and we compute the moments of the limit distribution.
