The lattice point counting problem on the Heisenberg groups
The lattice point counting problem on the Heisenberg groups
复制标题
海森堡群上的格点计数问题
DOI：
10.5802/aif.2986
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
K. Taylor
中科院分区：
文献类型：
--
作者：
Rahul Garg;A. Nevo;K. Taylor
我们考虑了Heisenberg群上的径向范数和Heisenberg齐次范数，它们由$N_{\alpha，A}（（z，t））= \left（|z| ^\alpha + A|不|^{\alpha/2}\right）^{1/\alpha}$，对于$\alpha \ge 2$和$A>0$。这个自然族包括规范的Cygan-Kor\'anyi范数，对应于$\alpha =4$。研究了Heisenberg群上的格点计数问题，即建立了大半径球R上整点格点个数的误差估计。我们在$\alpha=2$的情况下为误差建立的指数是所有维度上的最佳可能。
We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by $N_{\alpha,A}((z,t)) = \left(|z|^\alpha + A |t|^{\alpha/2}\right)^{1/\alpha}$, for $\alpha \ge 2$ and $A>0$. This natural family includes the canonical Cygan-Kor\'anyi norm, corresponding to $\alpha =4$. We study the lattice points counting problem on the Heisenberg groups, namely establish an error estimate for the number of points that the lattice of integral points has in a ball of large radius $R$. The exponent we establish for the error in the case $\alpha=2$ is the best possible, in all dimensions.