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Lattice Points Close to the Heisenberg Spheres

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Abstract

We study a lattice point-counting problem for spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic norms \(\Vert (z,t)\Vert _\alpha = ( \left| z\right| ^\alpha + \left| t\right| ^{\alpha /2})^{1/\alpha }\) for \(\alpha \ge 2\). As a first step, we reduce our counting problem to one of bounding an energy integral. The primary new challenges that arise are the presence of vanishing curvature and nonisotropic dilations. In the process, we establish bounds on the Fourier transform of the surface measures arising from these norms. Further, we utilize the techniques developed here to estimate the number of lattice points in the intersection of two such surfaces.

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Notes

  1. Here, \(x \lesssim y\) means that there exists some constant C such that \(x \le Cy\), and \(x \sim y\) means that both \(x \lesssim y\) and \(y \lesssim x\).

  2. Technically, given \(({\tilde{z}},z_n) \in {\mathbb {R}}^n\), \(z_n-\varphi ({\tilde{z}})\) is the defining function. A limited working explanation of this process is provided here; for the full details of these terms and relations see [37, Chapter 8].

  3. In the boundary case, where \(r_0=\epsilon \), the first integral is simply zero and the second becomes an integral to \(2\epsilon \), which produces no substantive change.

  4. In the boundary case, where \(r_0=|\xi _3|^{-\frac{1}{\alpha }}\), so \(r_0=\epsilon \), the first integral is simply zero and the second becomes an integral to \(2\epsilon \), which produces no substantive change.

  5. For \(\alpha =4\) and \(d \ge 3\), Gath’s result in [14] would provide a similar improvement.

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Acknowledgements

We would like to acknowledge Professor Allan Greenleaf at the University of Rochester for sharing invaluable insights and feedback that greatly improved our article. Thank you for your endless patience and kindness.

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Correspondence to Krystal Taylor.

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Taylor is supported in part by the Simons Foundation Grant 523555.

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Campolongo, E.G., Taylor, K. Lattice Points Close to the Heisenberg Spheres. La Matematica 2, 156–196 (2023). https://doi.org/10.1007/s44007-022-00040-z

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