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Counting Lattice Points near Korányi Spheres via Generalized Radon Transforms

Rajula Srivastava, Krystal Taylor

TL;DR

...

Abstract

In this note, we study a lattice point counting problem for spheres in Heisenberg groups, incorporating both the non-isotropic dilation structure and the non-commutative group law. More specifically, we establish an upper bound for the average number of lattice points in a $δ$-neighborhood of a Korányi sphere of large radius, where the average considered is over Heisenberg group translations of the sphere. This is in contrast with previous works, which either count lattice points on dilates of a fixed sphere (see \cites{GNT, Gath2}) or consider averages over Euclidean translations of the sphere (see \cites{CT}). We observe that incorporating the Heisenberg group structure allows us to circumvent the degeneracy arising from the vanishing of the Gaussian curvature at the poles of the Korányi sphere. In fact, in lower dimensions (the first and second Heisenberg group), our method establishes an upper bound for this number which gives a logarithmic improvement over the bound implied by the previously known results. Even for the higher dimensional Heisenberg groups, we recover the bounds implied by the main result of \cite{GNT} using a completely different approach of generalized Radon transforms. Further, we obtain upper bounds for the average number of lattice points near more general spheres described with respect to radial, Heisenberg homogeneous norms as considered in \cite{GNT}.

Counting Lattice Points near Korányi Spheres via Generalized Radon Transforms

TL;DR

...

Abstract

In this note, we study a lattice point counting problem for spheres in Heisenberg groups, incorporating both the non-isotropic dilation structure and the non-commutative group law. More specifically, we establish an upper bound for the average number of lattice points in a -neighborhood of a Korányi sphere of large radius, where the average considered is over Heisenberg group translations of the sphere. This is in contrast with previous works, which either count lattice points on dilates of a fixed sphere (see \cites{GNT, Gath2}) or consider averages over Euclidean translations of the sphere (see \cites{CT}). We observe that incorporating the Heisenberg group structure allows us to circumvent the degeneracy arising from the vanishing of the Gaussian curvature at the poles of the Korányi sphere. In fact, in lower dimensions (the first and second Heisenberg group), our method establishes an upper bound for this number which gives a logarithmic improvement over the bound implied by the previously known results. Even for the higher dimensional Heisenberg groups, we recover the bounds implied by the main result of \cite{GNT} using a completely different approach of generalized Radon transforms. Further, we obtain upper bounds for the average number of lattice points near more general spheres described with respect to radial, Heisenberg homogeneous norms as considered in \cite{GNT}.

Paper Structure

This paper contains 23 sections, 12 theorems, 152 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main Results
  3. Background and Previous Work
  4. The Euclidean setting
  5. The variable coefficient, multi-parameter setting
  6. Lattice point counting on Heisenberg groups
  7. Comparison with Previous Results
  8. The trivial bound
  9. Results on Heisenberg Lattice Point Counting Problems
  10. Structure of the Paper
  11. Notation
  12. Acknowledgement
  13. Proof of Theorems \ref{['maingeneral']} and \ref{['maingeneral not4']}
  14. Main Counting Lemma
  15. Proof of Theorems \ref{['maingeneral']} and \ref{['maingeneral not4']} assuming Lemma \ref{['main lemma']}
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $n\geq 1$ and set $D=2n+1$. For $\delta\in (0,1)$, $Q\in \mathbb{N}$, and $\phi_4$ as in phi (with $\alpha=4)$, we have Here the implicit constants depend only on $\phi_\alpha$ and $D$.

Figures (4)

  • Figure 1: $x=(0,0,0)$, $y_2=x_2$
  • Figure 2: $x=(3,0,3)$, $y_2=x_2$
  • Figure 3: $x=(3,2,2)$, $y_2=x_2$
  • Figure 4: The sets $\tau_q^a(E_q)$ (left) and the scaling to $E_q$ (right), where $\tau_q^a$ is the scaling operation in \ref{['scalingop']}.

Theorems & Definitions (21)

  • Theorem 1.1: Main result for the Korányi sphere $\alpha=4$
  • Theorem 1.2: Main result $\alpha\neq 4$
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1: Main Counting Lemma
  • Definition 2.2: Probability measure on $E_q$
  • Lemma 2.3: transforming the lattice point counting problem to a measure-theoretic bound
  • proof
  • ...and 11 more