海森堡群上 Cygan - Korányi 球中的格点计数
Lattice point counting in Cygan--Korányi balls on Heisenberg groups
浏览论文内容
中文总结 AI 辅助
本文通过Landau公式和van der Corput导数测试，改进了海森堡群上Cygan-Korányi球体格点计数的误差界，为Gath猜想提供了新的进展。
中文摘要 AI 辅助
海森堡群\(\mathbb{H}^q\)上规范球中的格点计数是欧几里得多维球问题的非交换类似问题，由 Garg、Nevo 和 Taylor 提出。特别有趣的情况是规范取为 Cygan - Korányi 范数，误差项\(\mathcal{E}_q(t)=\#\left(\mathbb{Z}^{2 q + 1} \cap \mathcal{B}_t\right)-\operatorname{vol}(\mathcal{B}_1) \, t^{2 q + 2}\)，\(\mathcal{B}_t=\{(v,w)\in\mathbb{H}^q: (|v|^4 + w^2)^{1/4} \le t \}\)，与高斯圆问题密切相关。当\(q\geq3\)时，Gath 改进了之前结果并提出猜想最优阶应为\(2q - 1\)。本文通过 Landau 公式和 van der Corput 的\(5,6\)阶导数测试，得出对于\(q \geq 4\)，\(|\mathcal{E}_q(t)| \lesssim t^{2 q - 1 + 241/753}\)，并在\(q = 3\)时恢复了 Gath 的界（至多一个对数因子），通过更简单方法为 Gath 的猜想取得了首个进展。
英文摘要
Lattice point counting in gauge balls on the Heisenberg group $\mathbb{H}^q$ is a non-commutative analogue of the Euclidean multidimensional sphere problem, initiated by Garg, Nevo and Taylor \cite[\textit{Ann. Inst. Fourier}, 2015]{GNT15}. The case of particular interest is when the gauge is taken as the Cygan--Korányi norm and the error term reads: $$\mathcal{E}_q(t)=\#\left(\mathbb{Z}^{2 q+1} \cap \mathcal{B}_t\right)-\operatorname{vol}(\mathcal{B}_1) \, t^{2 q+2},$$ with $\mathcal{B}_t=\{(v,w)\in\mathbb{H}^q: (|v|^4 + w^2)^{1/4} \le t \}$, which is closely related to the Gauss circle problem. When $q\ge3$, Gath \cite[\textit{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 2022]{Gat22} improved upon \cite[]{GNT15} by showing that $ |\mathcal{E}_q(t)|\lesssim t^{2q-1+ 1/3}$ and proposed the conjecture that the optimal order should be $2q-1$. In this paper, through Landau's formula and the $5,6$-th Derivative Tests of van der Corput, we arrive at that $|\mathcal{E}_q(t)| \lesssim t^{2 q-1 + 241/753} $ for any $ q \geq 4$, and recover the bound of Gath for $q=3$ up to a logarithmic factor. This, via a simpler method, provides the first progress towards Gath's conjecture.
