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

Garg, Rahul; Nevo, Amos; Taylor, Krystal

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

Abstract
Résumé

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.

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.

Received: 2014-04-24
Revised: 2014-09-14
Accepted: 2014-11-27
Published online: 2015-11-09

MR: 3449210 | Zbl: 1358.52017

DOI: 10.5802/aif.2986
Classification: 11P21, 43A80, 42B99, 26D10
Keywords: Heisenberg groups, lattice points, Poisson summation formula, Cygan-Koranyi norm

@article{AIF_2015__65_5_2199_0,
     author = {Garg, Rahul and Nevo, Amos and Taylor, Krystal},
     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},
     volume = {65},
     number = {5},
     year = {2015},
     doi = {10.5802/aif.2986},
     zbl = {1358.52017},
     mrnumber = {3449210},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2986/}
}

TY  - JOUR
TI  - The lattice point counting problem  on the Heisenberg groups
JO  - Annales de l'Institut Fourier
PY  - 2015
DA  - 2015///
SP  - 2199
EP  - 2233
VL  - 65
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2986/
UR  - https://zbmath.org/?q=an%3A1358.52017
UR  - https://www.ams.org/mathscinet-getitem?mr=3449210
UR  - https://doi.org/10.5802/aif.2986
DO  - 10.5802/aif.2986
LA  - en
ID  - AIF_2015__65_5_2199_0
ER  -

%0 Journal Article
%T The lattice point counting problem  on the Heisenberg groups
%J Annales de l'Institut Fourier
%D 2015
%P 2199-2233
%V 65
%N 5
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2986
%R 10.5802/aif.2986
%G en
%F AIF_2015__65_5_2199_0

Garg, Rahul; Nevo, Amos; Taylor, Krystal. The lattice point counting problem  on the Heisenberg groups. Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2199-2233. doi : 10.5802/aif.2986. https://aif.centre-mersenne.org/articles/10.5802/aif.2986/

References
Cited by

[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, Tome 110 (2013) no. 48, pp. 19220-19226 | Article | MR: 3153949 | Zbl: 1294.53041

[2] Chamizo, Fernando Lattice points in bodies of revolution, Acta Arith., Tome 85 (1998) no. 3, pp. 265-277 | EuDML: 207168 | MR: 1627839 | Zbl: 0919.11061

[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: 777340

[4] Cowling, Michael; Dooley, Anthony H.; Korányi, Adam; Ricci, Fulvio $H$ -type groups and Iwasawa decompositions, Adv. Math., Tome 87 (1991) no. 1, pp. 1-41 | Article | MR: 1102963 | Zbl: 0761.22010

[5] Cygan, Jacek Wiener’s test for the Brownian motion on the Heisenberg group, Colloq. Math., Tome 39 (1978) no. 2, pp. 367-373 | EuDML: 263661 | MR: 522380 | Zbl: 0409.60075

[6] Cygan, Jacek Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc., Tome 83 (1981) no. 1, p. 69-70 | Article | MR: 619983 | Zbl: 0475.43010

[7] Duchin, Moon; Mooney, Christopher Fine asymptotic geometry in the Heisenberg group, Indiana Univ. Math. J., Tome 63 (2014) no. 3, pp. 885-916 | Article | MR: 3254527 | Zbl: 1346.20055

[8] Erdélyi, A. Asymptotic expansions, Dover Publications, Inc., New York, 1956, vi+108 pages | MR: 78494 | Zbl: 0070.29002

[9] Frank, Rupert L.; Lieb, Elliott H. Sharp constants in several inequalities on the Heisenberg group, Ann. of Math. (2), Tome 176 (2012) no. 1, pp. 349-381 | Article | MR: 2925386 | Zbl: 1252.42023

[10] Gorodnik, Alexander; Nevo, Amos Counting lattice points, J. Reine Angew. Math., Tome 663 (2012), pp. 127-176 | Article | Zbl: 1248.37011

[11] Grafakos, Loukas Classical Fourier analysis, Graduate Texts in Mathematics, Tome 249, Springer, New York, 2008, xvi+489 pages | Zbl: 1220.42001

[12] Herz, C. S. On the number of lattice points in a convex set, Amer. J. Math., Tome 84 (1962), pp. 126-133 | Zbl: 0113.03703

[13] Hlawka, Edmund Über Integrale auf konvexen Körpern. I, Monatsh. Math., Tome 54 (1950), pp. 1-36 | Zbl: 0036.30902

[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: 1177.11084

[15] Khosravi, Mahta; Petridis, Yiannis N. The remainder in Weyl’s law for $n$ -dimensional Heisenberg manifolds, Proc. Amer. Math. Soc., Tome 133 (2005) no. 12, p. 3561-3571 (electronic) | Article | Zbl: 1080.35054

[16] Korányi, Adam Geometric properties of Heisenberg-type groups, Adv. in Math., Tome 56 (1985) no. 1, pp. 28-38 | Article | Zbl: 0589.53053

[17] Krätzel, Ekkehard Lattice points, Mathematics and its Applications (East European Series), Tome 33, Kluwer Academic Publishers Group, Dordrecht, 1988, 320 pages | Zbl: 0675.10031

[18] Krätzel, Ekkehard Lattice points in super spheres, Comment. Math. Univ. Carolin., Tome 40 (1999) no. 2, pp. 373-391 | Zbl: 0993.11050

[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., Tome 43 (2002) no. 4, pp. 755-771 | Zbl: 1064.11064

[20] Krätzel, Ekkehard Lattice points in three-dimensional convex bodies with points of Gaussian curvature zero at the boundary, Monatsh. Math., Tome 137 (2002) no. 3, pp. 197-211 | Article | Zbl: 1016.11045

[21] Krätzel, Ekkehard; Nowak, Werner Georg The lattice discrepancy of bodies bounded by a rotating Lamé’s curve, Monatsh. Math., Tome 154 (2008) no. 2, pp. 145-156 | Article | Zbl: 1253.11092

[22] Krätzel, Ekkehard; Nowak, Werner Georg The lattice discrepancy of certain three-dimensional bodies, Monatsh. Math., Tome 163 (2011) no. 2, pp. 149-174 | Article | Zbl: 1258.11087

[23] Nowak, Werner Georg On the lattice discrepancy of bodies of rotation with boundary points of curvature zero, Arch. Math. (Basel), Tome 90 (2008) no. 2, pp. 181-192 | Article | Zbl: 1139.11042

[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., Tome 233 (2000) no. 4, pp. 803-815 | Article | Zbl: 1125.11343

[26] Peter, Manfred Lattice points in convex bodies with planar points on the boundary, Monatsh. Math., Tome 135 (2002) no. 1, pp. 37-57 | Article | Zbl: 1055.11059

[27] Randol, Burton A lattice-point problem, Trans. Amer. Math. Soc., Tome 121 (1966), pp. 257-268 | Zbl: 0135.10601

[28] Randol, Burton A lattice-point problem. II, Trans. Amer. Math. Soc., Tome 125 (1966), pp. 101-113 | Zbl: 0161.04902

[29] Randol, Burton On the asymptotic behavior of the Fourier transform of the indicator function of a convex set, Trans. Amer. Math. Soc., Tome 139 (1969), pp. 279-285 | Zbl: 0183.26905

[30] Randol, Burton On the Fourier transform of the indicator function of a planar set., Trans. Amer. Math. Soc., Tome 139 (1969), pp. 271-278 | Zbl: 0183.26904

[31] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, Tome 43, Princeton University Press, Princeton, NJ, 1993, xiv+695 pages (With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III) | Zbl: 0821.42001

[32] Stein, Elias M.; Wainger, Stephen Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc., Tome 84 (1978) no. 6, pp. 1239-1295 | Article | Zbl: 0393.42010

Cited by Sources:

ANNALES DE L'INSTITUT FOURIER

ARTICLES TO APPEAR

BROWSE ISSUES

ABOUT THE JOURNAL

EDITORIAL BOARD

SUBMIT A PAPER

SUBSCRIPTION