论文标题
限制晶格问题中的法律。 V.分析和精确凸集的情况
Limit laws in the lattice problem. V. The case of analytic and stricly convex sets
论文作者
论文摘要
我们研究了一个具有原始凸面和分析集中的单模型晶格的点数的误差，并由因子$ t $扩张。目的是概括上一篇文章的结果。我们首先表明，当该错误通过$ \ sqrt {t} $归一化时，当此参数趋于无限时，并且当考虑晶格是随机的时，将其减少到研究siegel transform $ \ mathcal $ \ nathcal {s}（f_ {t}）（f _ {t}）（l）$，这些取决于$ t $。然后，我们回到研究带有随机权重的Siegel变换的渐近行为，$ \ Mathcal {s}（f）（θ，l）$，其中$θ$是第二个随机参数。然后，我们表明，最后数量几乎肯定会融合，我们研究了其定律时刻的存在。最后，我们表明，如果我们在\ at \ mathbb {r}^{2} $中翻译固定向量$α\的严格凸集，则此结果仍然有效。
We study the error of the number of points of a unimodular lattice that fall in a strictly convex and analytic set having the origin and that is dilated by a factor $t$. The aim is to generalize the result of a previous article. We first show that the study of the error, when it is normalized by $\sqrt{t}$, when this parameter tends to infinity and when the considered lattice is random, is reduced to the study of a Siegel transform $\mathcal{S}(f_{t})(L)$ which depends on $t$. Then, we come back to the study of the asymptotic behaviour of a Siegel transform with random weights, $\mathcal{S}(F)(θ,L)$ where $θ$ is a second random parameter. Then, we show that this last quantity converges almost surely and we study the existence of moments of its law. Finally, we show that this result is still valid if we translate, after dilation, the strictly convex set of a fixed vector $α\in \mathbb{R}^{2}$.
