Lattice point counting problems on step-two nilpotent Lie groups

Technical Summary: Lattice Point Counting on Step-Two Nilpotent Lie Groups

Problem Statement
This paper addresses the lattice point counting problem on connected and simply connected nilpotent Lie groups of step-two, denoted as $G(q, m)$. The specific objective is to estimate the discrepancy between the number of lattice points contained in a dilated ball and the volume of that ball as the radius $R \to \infty$.

The domains considered are balls $B^{\alpha, M}_R$ associated with a family of homogeneous norms $N_{\alpha, M}(x, t) = (|M_1 x|^\alpha + |M_2 t|^{\alpha/2})^{1/\alpha}$, where $(x, t) \in \mathbb{R}^q \times \mathbb{R}^m$. These norms are adapted to the parabolic dilation structure $\delta_r(x, t) = (rx, r^2t)$. The study covers all dimensions $q \ge 2$ and $m \ge 1$, and all exponents $\alpha > 0$. The lattices $\Gamma_L$ are defined via invertible matrices $L$, encompassing standard integer lattices and more general discrete subgroups.

The core difficulty lies in the geometry of the domain boundaries. Unlike Euclidean $\ell_\alpha$-balls, the unit spheres $\partial B^{\alpha, M}_1$ in these groups lack rotational symmetry and exhibit anisotropy. Furthermore, the curvature properties vary significantly with $\alpha$: for $\alpha < 4$, the boundary is not $C^2$; for $\alpha \ge 4$, the Gaussian curvature vanishes on the equator ($t=0$) and the pole ($x=0$), leading to degenerate stationary points in the associated oscillatory integrals.

Methodology
The authors employ a combination of harmonic analysis and asymptotic analysis techniques:

1. Poisson Summation: The counting problem is transformed into a spectral problem using the Poisson summation formula. The discrepancy is bounded by the sum of the Fourier transforms of the characteristic functions of the balls, excluding the zero frequency.
2. Spectral Estimates: The primary technical effort focuses on deriving sharp decay estimates for the Fourier transform $\widehat{\chi_{B^{\alpha, M}_1}}(w, s)$ of the unit ball's characteristic function.
3. Oscillatory Integral Analysis: The Fourier transform is expressed as an integral involving Bessel functions. The authors analyze these integrals by:
   • Utilizing the recursion formulas of Bessel functions to perform integration by parts, thereby exploiting cancellation and improving decay rates.
   • Applying Van der Corput lemmas to estimate oscillatory integrals with singular phase functions.
   • Conducting a resolution of the identity to separate the integration domain into regions where different asymptotic behaviors of the Bessel functions dominate.
4. Anisotropic Analysis: A key methodological innovation is the distinction between two regimes based on the relative sizes of the frequency parameters $\lambda_1 \sim |w|$ and $\lambda_2 \sim |s|$. The authors treat the cases $\lambda_1 \gtrsim \lambda_2$ and $\lambda_1 \lesssim \lambda_2$ separately, utilizing different integral representations (involving different orders of Bessel functions) for each case to handle the anisotropy of the group structure.
5. Subadditivity Arguments: For $\alpha \in (0, 1]$, where the norm $N_{\alpha, M}$ is not subadditive, the authors utilize the subadditivity of $N_{\alpha, M}^\alpha$ to construct appropriate smooth approximations for the counting function.

Key Contributions and Results

• Generalization: The theory is extended from the Heisenberg group ($m=1$) to general step-two nilpotent groups with arbitrary central dimension $m$.
• Explicit Discrepancy Bounds: The paper provides explicit upper bounds for the relative discrepancy $P^{\alpha, M}_{q, m}(R)$ for all $\alpha > 0$ and all dimensions. The bounds are expressed in terms of powers of $R$ and logarithmic factors, depending on the relationship between $\alpha, q,$ and $m$.
• Quantitative Improvements on Heisenberg Groups: The results quantitatively improve upon previous work by Garg, Nevo, and Taylor [17] on Heisenberg groups ($H_d$, where $q=2d, m=1$):
  • Dimension 5 ($d=2$): For $\alpha \in (3, 4)$ or $\alpha = 1$, the exponent of the logarithmic factor is reduced from $2/3$ to $1/3$. For $\alpha = 4$ (Cygan–Korányi norm) and $\alpha \in (2, 3]$, the logarithmic factor $\log^{2/3} R$ is completely removed.
  • Dimension 3 ($d=1$): For $\alpha = 1$, the estimate is upgraded from $O_\epsilon(R^{5/2+\epsilon})$ to $O(R^2 \log^{1/2} R)$. For $\alpha \in (1, 2)$, the bound is improved to $O(R^{19/8})$. For $\alpha > 4$, the logarithmic factor $\log R$ is removed.
• Sharpness: The bounds are shown to be sharp in the case where the center is unidimensional ($m=1$) and $\alpha = 2$, provided the matrix parameters satisfy a rationality condition.
• Shell Counting: As a byproduct, the authors extend results on lattice point counting near hypersurfaces (spheres) to the general step-two setting, providing estimates for the number of points in thin shells $\partial B^{\alpha, M}_R$. This improves upon recent results by Campolongo–Taylor [8] and Srivastava–Taylor [58] for Heisenberg groups.

Significance and Claims
The paper claims to provide a "complete understanding" of the distribution of lattice points within a non-Euclidean frame for step-two nilpotent groups, achieving optimal discrepancy exponents in several cases.

The authors emphasize that their method overcomes limitations of previous approaches:

• Unlike the slicing arguments used by Gath [18, 19], which are restricted to specific arithmetic structures (like $\alpha=4$), the current method applies to general $\alpha$ and general matrices.
• Unlike the methods of Garg, Nevo, and Taylor [17], which struggle with $\alpha < 2$ due to non-convexity and degenerate stationary points, the current approach utilizes refined Bessel function analysis and phase function decomposition to handle these regimes.
• The work addresses the "anisotropy" of the group structure, a feature not fully captured in prior Heisenberg-specific frameworks, by distinguishing between frequency regimes $\lambda_1$ and $\lambda_2$.

The study is presented as a preliminary step toward a uniform understanding of discrepancy estimates in $q$ and $m$, with the authors noting that further optimization for large dimensions remains a subject for future work. The results are derived using standard tools of analytic number theory and harmonic analysis (Poisson summation, oscillatory integrals, Bessel functions) adapted to the specific geometry of step-two nilpotent Lie groups.