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.

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

[1] Breuillard, Emmanuel; Le Donne, Enrico On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry, Proc. Natl. Acad. Sci. USA, Volume 110 (2013) no. 48, pp. 19220-19226 | DOI | MR | Zbl

[2] Chamizo, Fernando Lattice points in bodies of revolution, Acta Arith., Volume 85 (1998) no. 3, pp. 265-277 | EuDML | MR | Zbl

[3] Cowling, Michael Unitary and uniformly bounded representations of some simple Lie groups, Harmonic analysis and group representations, Liguori, Naples, 1982, pp. 49-128 | MR

[4] Cowling, Michael; Dooley, Anthony H.; Korányi, Adam; Ricci, Fulvio $H$-type groups and Iwasawa decompositions, Adv. Math., Volume 87 (1991) no. 1, pp. 1-41 | DOI | MR | Zbl

[5] Cygan, Jacek Wiener’s test for the Brownian motion on the Heisenberg group, Colloq. Math., Volume 39 (1978) no. 2, pp. 367-373 | EuDML | MR | Zbl

[6] Cygan, Jacek Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc., Volume 83 (1981) no. 1, pp. 69-70 | DOI | MR | Zbl

[7] Duchin, Moon; Mooney, Christopher Fine asymptotic geometry in the Heisenberg group, Indiana Univ. Math. J., Volume 63 (2014) no. 3, pp. 885-916 | DOI | MR | Zbl

[8] Erdélyi, A. Asymptotic expansions, Dover Publications, Inc., New York, 1956, pp. vi+108 | MR | Zbl

[9] Frank, Rupert L.; Lieb, Elliott H. Sharp constants in several inequalities on the Heisenberg group, Ann. of Math. (2), Volume 176 (2012) no. 1, pp. 349-381 | DOI | MR | Zbl

[10] Gorodnik, Alexander; Nevo, Amos Counting lattice points, J. Reine Angew. Math., Volume 663 (2012), pp. 127-176 | DOI | Zbl

[11] Grafakos, Loukas Classical Fourier analysis, Graduate Texts in Mathematics, 249, Springer, New York, 2008, pp. xvi+489 | Zbl

[12] Herz, C. S. On the number of lattice points in a convex set, Amer. J. Math., Volume 84 (1962), pp. 126-133 | Zbl

[13] Hlawka, Edmund Über Integrale auf konvexen Körpern. I, Monatsh. Math., Volume 54 (1950), pp. 1-36 | Zbl

[14] Ivić, A.; Krätzel, E.; Kühleitner, M.; Nowak, W. G. Lattice points in large regions and related arithmetic functions: recent developments in a very classic topic, Elementare und analytische Zahlentheorie (Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20), Franz Steiner Verlag Stuttgart, Stuttgart, 2006, pp. 89-128 | Zbl

[15] Khosravi, Mahta; Petridis, Yiannis N. The remainder in Weyl’s law for $n$-dimensional Heisenberg manifolds, Proc. Amer. Math. Soc., Volume 133 (2005) no. 12, p. 3561-3571 (electronic) | DOI | Zbl

[16] Korányi, Adam Geometric properties of Heisenberg-type groups, Adv. in Math., Volume 56 (1985) no. 1, pp. 28-38 | DOI | Zbl

[17] Krätzel, Ekkehard Lattice points, Mathematics and its Applications (East European Series), 33, Kluwer Academic Publishers Group, Dordrecht, 1988, pp. 320 | Zbl

[18] Krätzel, Ekkehard Lattice points in super spheres, Comment. Math. Univ. Carolin., Volume 40 (1999) no. 2, pp. 373-391 | Zbl

[19] Krätzel, Ekkehard Lattice points in some special three-dimensional convex bodies with points of Gaussian curvature zero at the boundary, Comment. Math. Univ. Carolin., Volume 43 (2002) no. 4, pp. 755-771 | Zbl

[20] Krätzel, Ekkehard Lattice points in three-dimensional convex bodies with points of Gaussian curvature zero at the boundary, Monatsh. Math., Volume 137 (2002) no. 3, pp. 197-211 | DOI | Zbl

[21] Krätzel, Ekkehard; Nowak, Werner Georg The lattice discrepancy of bodies bounded by a rotating Lamé’s curve, Monatsh. Math., Volume 154 (2008) no. 2, pp. 145-156 | DOI | Zbl

[22] Krätzel, Ekkehard; Nowak, Werner Georg The lattice discrepancy of certain three-dimensional bodies, Monatsh. Math., Volume 163 (2011) no. 2, pp. 149-174 | DOI | Zbl

[23] Nowak, Werner Georg On the lattice discrepancy of bodies of rotation with boundary points of curvature zero, Arch. Math. (Basel), Volume 90 (2008) no. 2, pp. 181-192 | DOI | Zbl

[24] Parkkonen, Jouni; Paulin, Frédéric Counting and equidistribution in Heisenberg groups (http://arxiv.org/abs/1402.7225)

[25] Peter, Manfred The local contribution of zeros of curvature to lattice point asymptotics, Math. Z., Volume 233 (2000) no. 4, pp. 803-815 | DOI | Zbl

[26] Peter, Manfred Lattice points in convex bodies with planar points on the boundary, Monatsh. Math., Volume 135 (2002) no. 1, pp. 37-57 | DOI | Zbl

[27] Randol, Burton A lattice-point problem, Trans. Amer. Math. Soc., Volume 121 (1966), pp. 257-268 | Zbl

[28] Randol, Burton A lattice-point problem. II, Trans. Amer. Math. Soc., Volume 125 (1966), pp. 101-113 | Zbl

[29] Randol, Burton On the asymptotic behavior of the Fourier transform of the indicator function of a convex set, Trans. Amer. Math. Soc., Volume 139 (1969), pp. 279-285 | Zbl

[30] Randol, Burton On the Fourier transform of the indicator function of a planar set., Trans. Amer. Math. Soc., Volume 139 (1969), pp. 271-278 | Zbl

[31] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993, pp. xiv+695 (With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III) | Zbl

[32] Stein, Elias M.; Wainger, Stephen Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc., Volume 84 (1978) no. 6, pp. 1239-1295 | DOI | Zbl