We consider radial and Heisenberg-homogeneous norms on the Heisenberg groups given by for and . This natural family includes the canonical Cygan-Korányi norm, corresponding to . 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 . The exponent we establish for the error in the case is the best possible, in all dimensions.
Nous considérons les normes radiales et Heisenberg-homogènes sur les groupes de Heisenberg données par , pour et . Cette famille naturelle inclut la norme canonique de Cygan-Korányi, qui correspond à . 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 . L’exposant utilisé pour le terme d’erreur dans le cas est optimal, en toute dimension.
Revised:
Accepted:
Published online:
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 -
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/
[1] 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] Lattice points in bodies of revolution, Acta Arith., Tome 85 (1998) no. 3, pp. 265-277 | EuDML: 207168 | MR: 1627839 | Zbl: 0919.11061
[3] Unitary and uniformly bounded representations of some simple Lie groups, Harmonic analysis and group representations, Liguori, Naples, 1982, pp. 49-128 | MR: 777340
[4] -type groups and Iwasawa decompositions, Adv. Math., Tome 87 (1991) no. 1, pp. 1-41 | Article | MR: 1102963 | Zbl: 0761.22010
[5] 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] 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] 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] Asymptotic expansions, Dover Publications, Inc., New York, 1956, vi+108 pages | MR: 78494 | Zbl: 0070.29002
[9] 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] Counting lattice points, J. Reine Angew. Math., Tome 663 (2012), pp. 127-176 | Article | Zbl: 1248.37011
[11] Classical Fourier analysis, Graduate Texts in Mathematics, Tome 249, Springer, New York, 2008, xvi+489 pages | Zbl: 1220.42001
[12] On the number of lattice points in a convex set, Amer. J. Math., Tome 84 (1962), pp. 126-133 | Zbl: 0113.03703
[13] Über Integrale auf konvexen Körpern. I, Monatsh. Math., Tome 54 (1950), pp. 1-36 | Zbl: 0036.30902
[14] 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] The remainder in Weyl’s law for -dimensional Heisenberg manifolds, Proc. Amer. Math. Soc., Tome 133 (2005) no. 12, p. 3561-3571 (electronic) | Article | Zbl: 1080.35054
[16] Geometric properties of Heisenberg-type groups, Adv. in Math., Tome 56 (1985) no. 1, pp. 28-38 | Article | Zbl: 0589.53053
[17] Lattice points, Mathematics and its Applications (East European Series), Tome 33, Kluwer Academic Publishers Group, Dordrecht, 1988, 320 pages | Zbl: 0675.10031
[18] Lattice points in super spheres, Comment. Math. Univ. Carolin., Tome 40 (1999) no. 2, pp. 373-391 | Zbl: 0993.11050
[19] 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] 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] 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] The lattice discrepancy of certain three-dimensional bodies, Monatsh. Math., Tome 163 (2011) no. 2, pp. 149-174 | Article | Zbl: 1258.11087
[23] 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] Counting and equidistribution in Heisenberg groups (http://arxiv.org/abs/1402.7225)
[25] 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] 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] A lattice-point problem, Trans. Amer. Math. Soc., Tome 121 (1966), pp. 257-268 | Zbl: 0135.10601
[28] A lattice-point problem. II, Trans. Amer. Math. Soc., Tome 125 (1966), pp. 101-113 | Zbl: 0161.04902
[29] 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] 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] 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] 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: