The lattice point counting problem  on the Heisenberg groups

Garg, Rahul; Nevo, Amos; Taylor, Krystal

doi:10.5802/aif.2986

The lattice point counting problem on the Heisenberg groups
[Le problème dénombrement des points d’un réseau dans les groupes de Heisenberg]

Garg, Rahul ¹ ; Nevo, Amos ¹ ; Taylor, Krystal ²

¹ Department of Mathematics Technion, Haifa-32000 (Israel)
² Institute for Mathematics and its Applications Minneapolis, Minnesota (USA)

Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2199-2233.

Résumé
Abstract

Nous considérons les normes radiales et Heisenberg-homogènes sur les groupes de Heisenberg données par $N_{α, A} ((z, t)) = {({|z|}^{α} + A {|t|}^{α / 2})}^{1 / α}$ , pour $α \geq 2$ et $A > 0$ . Cette famille naturelle inclut la norme canonique de Cygan-Korányi, qui correspond à $α = 4$ . Nous étudions le problème de dénombrement des points d’un réseau dans les groupes de Heisenberg, et nous établissons un terme d’erreur sur le nombre d’éléments du réseau des points entiers dans une boule de grand rayon $R$ . L’exposant utilisé pour le terme d’erreur dans le cas $α = 2$ est optimal, en toute dimension.

We consider radial and Heisenberg-homogeneous norms on the Heisenberg groups given by $N_{α, A} ((z, t)) = {({|z|}^{α} + A {|t|}^{α / 2})}^{1 / α},$ for $α \geq 2$ and $A > 0$ . This natural family includes the canonical Cygan-Korányi norm, corresponding to $α = 4$ . 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 $α = 2$ is the best possible, in all dimensions.

MR Zbl

DOI : 10.5802/aif.2986

Classification : 11P21, 43A80, 42B99, 26D10
Keywords: Heisenberg groups, lattice points, Poisson summation formula, Cygan-Koranyi norm
Mot clés : Groupes de Heisenberg, réseau de points, formule de sommes de Poisson, norme de Cygan-Koranyi

Affiliations des auteurs :

Garg, Rahul ¹ ; Nevo, Amos ¹ ; Taylor, Krystal ²

¹ Department of Mathematics Technion, Haifa-32000 (Israel)
² Institute for Mathematics and its Applications Minneapolis, Minnesota (USA)

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     title = {The lattice point counting problem  on the {Heisenberg} groups},
     journal = {Annales de l'Institut Fourier},
     pages = {2199--2233},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Garg, Rahul; Nevo, Amos; Taylor, Krystal. The lattice point counting problem  on the Heisenberg groups. Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2199-2233. doi : 10.5802/aif.2986. https://aif.centre-mersenne.org/articles/10.5802/aif.2986/

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