论文标题
关于海森堡小组标准球中晶格点数的分布
On the Distribution of the Number of Lattice Points in Norm Balls on the Heisenberg Groups
论文作者
论文摘要
我们研究了位于cygan-kor {Á} nyi Norm Ball of ligh Radius的Heisenberg群体上积分晶格点的波动。令$ \ MATHCAL { Heisenberg Group $ \ Mathbb {H} _ {Q} $上的这个晶格计数问题，其中$ \ Mathcal {B} $是Cygan-kor {á} Nyi Norm和$δ_{X} $的单位球，是Heisenberg-Diles the Heisenberg-Diles by $ x> 0 $ x> 0 $ x> 0 $。对于$ q \ geq3 $，我们考虑适当的归一化错误项$ \ MATHCAL {e} _ {q}（x）/x^{2q-1} $，并证明其具有限制值分布，与Lebesgue Measure相对于Lebesgue Measure绝对连续。我们表明，该分布的定义密度，由$ \ Mathcal {p} _ {q}（α）$表示，可以将其扩展到整个复杂平面$ \ Mathbb {c} $作为$α$的整个函数，并满足任何非nengative integer integer integer integer $ j \ geq0 $ and $ j \ geq0 $ and $α$} $ |α|>α_{q，j} $，bond：\ begin {equination*} \ big | \ Mathcal {P}^{（J）} _ {q}（α）\ big | \ leq \ leq \ exp {\ big（ - | |α|^{4-β/\ log \ log \ log \ log \ log \ log {|α|α| axpect）此外，我们为任何整数$ j \ geq1 $的密度$ \ MATHCAL {p} _ {q} _ {q} _ {q} $的$ j $ -th积分矩提供明确的公式。
We investigate the fluctuations in the number of integral lattice points on the Heisenberg groups which lie inside a Cygan-Kor{á}nyi norm ball of large radius. Let $\mathcal{E}_{q}(x)=\big|\mathbb{Z}^{2q+1}\capδ_{x}\mathcal{B}\big|-\textit{vol}\big(\mathcal{B}\big)x^{2q+2}$ denote the error term which occurs for this lattice point counting problem on the Heisenberg group $\mathbb{H}_{q}$, where $\mathcal{B}$ is the unit ball in the Cygan-Kor{á}nyi norm and $δ_{x}$ is the Heisenberg-dilation by $x>0$. For $q\geq3$ we consider the suitably normalized error term $\mathcal{E}_{q}(x)/x^{2q-1}$, and prove it has a limiting value distribution which is absolutely continuous with respect to the Lebesgue measure. We show that the defining density for this distribution, denoted by $\mathcal{P}_{q}(α)$, can be extended to the whole complex plane $\mathbb{C}$ as an entire function of $α$ and satisfies for any non-negative integer $j\geq0$ and any $α\in\mathbb{R}$, $|α|>α_{q,j}$, the bound: \begin{equation*} \begin{split} \big|\mathcal{P}^{(j)}_{q}(α)\big|\leq\exp{\Big(-|α|^{4-β/\log\log{|α|}}\Big)} {split} {equation*} where $β>0$ is an absolute constant. In addition, we give an explicit formula for the $j$-th integral moment of the density $\mathcal{P}_{q}(α)$ for any integer $j\geq1$.
