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- W4295163683 abstract "We investigate the fluctuations in the number of integral lattice points on the Heisenberg groups which lie inside a Cygan-Kor{'a}nyi norm ball of large radius. Let $mathcal{E}_{q}(x)=big|mathbb{Z}^{2q+1}capdelta_{x}mathcal{B}big|-textit{vol}big(mathcal{B}big)x^{2q+2}$ 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{'a}nyi norm and $delta_{x}$ is the Heisenberg-dilation by $x>0$. For $qgeq3$ 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 $mathcal{P}_{q}(alpha)$, can be extended to the whole complex plane $mathbb{C}$ as an entire function of $alpha$ and satisfies for any non-negative integer $jgeq0$ and any $alphainmathbb{R}$, $|alpha|>alpha_{q,j}$, the bound: begin{equation*} begin{split} big|mathcal{P}^{(j)}_{q}(alpha)big|leqexp{Big(-|alpha|^{4-beta/loglog{|alpha|}}Big)} {split} {equation*} where $beta>0$ is an absolute constant. In addition, we give an explicit formula for the $j$-th integral moment of the density $mathcal{P}_{q}(alpha)$ for any integer $jgeq1$." @default.
- W4295163683 created "2022-09-11" @default.
- W4295163683 creator A5025171303 @default.
- W4295163683 date "2020-10-02" @default.
- W4295163683 modified "2023-10-17" @default.
- W4295163683 title "On the Distribution of the Number of Lattice Points in Norm Balls on the Heisenberg Groups" @default.
- W4295163683 doi "https://doi.org/10.48550/arxiv.2010.01096" @default.
- W4295163683 hasPublicationYear "2020" @default.
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