The lattice point counting problem on the Heisenberg groups

Garg, Rahul; Nevo, Amos; Taylor, Krystal

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

Nous considérons les normes radiales et Heisenberg-homogènes sur les groupes de Heisenberg données par $N_{\alpha ,A}\left((z,t)\right)={\left({\left|z\right|}^{\alpha }+A{\left|t\right|}^{\alpha /2}\right)}^{1/\alpha }$, pour $\alpha \ge 2$ et $A>0$. Cette famille naturelle inclut la norme canonique de Cygan-Korányi, qui correspond à $\alpha =4$. 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 $\alpha =2$ est optimal, en toute dimension.

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

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