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Distribution of shapes of orthogonal lattices. (English) Zbl 1412.37004

Given a unimodoular lattice (discrete subgroup of covolume 1) in \(\mathbb R^d\), one can consider the distribution of the directions of lattice points, that is, the projection of (primitive) lattice points to the unit sphere \(S^{d-1}\). If we consider the points of norm at most \(T\), we obtain a finitely supported measure on \(S^{d-1}\), and we can ask about its limits as \(T \rightarrow \infty\). A more refined question is to ask about the joint distribution of the direction of vectors together with the projection of the lattice to the orthogonal complement, a lattice in one lower dimension. The paper gives a general effective proof of a joint equidistribution result for this question, using effective equidistribution results for unipotent flows and removing some congruence conditions that were needed in dimensions \(d=3, 4, 5\) in prior works.

MSC:

37A17 Homogeneous flows
11H06 Lattices and convex bodies (number-theoretic aspects)
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[1] Aka, M., Einsiedler, M. and Shapira, U.. Integer points on spheres and their orthogonal grids. J. Lond. Math. Soc. (2)93(1) (2016), 143-158. · Zbl 1355.37016
[2] Aka, M., Einsiedler, M. and Shapira, U.. Integer points on spheres and their orthogonal lattices. Invent. Math.206(2) (2016), 379-396. · Zbl 1410.11117
[3] Borel, A. and Prasad, G.. Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups. Publ. Math. Inst. Hautes Études Sci.69 (1989), 119-171. · Zbl 0707.11032
[4] Dani, S. and Margulis, G.. Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci.101(1) (1991), 1-17. · Zbl 0731.22008
[5] Duke, W., Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math., 92, 1, 73-90, (1988) · Zbl 0628.10029
[6] Einsiedler, M., Margulis, G., Mohammadi, A. and Venkatesh, A.. Effective equidistribution and property (tau). Preprint, 2015 arXiv:1503.05884, submitted. · Zbl 1466.11049
[7] Einsiedler, M., Mozes, S., Shah, N. and Shapira, U.. Equidistribution of primitive rational points on expanding horospheres. Compos. Math.152(4) (2015), 667-692. · Zbl 1382.37032
[8] Einsiedler, M., Margulis, G. and Venkatesh, A.. Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces. Invent. Math.177(1) (2009), 137-212. · Zbl 1176.37003
[9] Ellenberg, J. S. and Venkatesh, A.. Local – global principles for representations of quadratic forms. Invent. Math.171(2) (2008), 257-279. · Zbl 1247.11048
[10] Gorodnik, A., Maucourant, F. and Oh, H.. Manin’s and Peyre’s conjectures on rational points and adelic mixing. Ann. Sci. Éc. Norm. Supér. (4)41(3) (2008), 383-435. · Zbl 1161.14015
[11] Gorodnik, A. and Oh, H.. Rational points on homogeneous varieties and equidistribution of adelic periods. Geom. Funct. Anal.21(2) (2011), 319-392. With an appendix by Mikhail Borovoi. · Zbl 1317.11069
[12] Hilbert, D., Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl nter Potenzen (Waringsches Problem), Math. Ann., 67, 3, 281-300, (1909) · JFM 40.0236.03
[13] Iwaniec, H., Fourier coefficients of modular forms of half-integral weight, Invent. Math., 87, 2, 385-401, (1987) · Zbl 0606.10017
[14] Kleinbock, D. Y. and Margulis, G. A.. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2)148(1) (1998), 339-360. · Zbl 0922.11061
[15] Kleinbock, D. and Tomanov, G.. Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation. Comment. Math. Helv.82(3) (2007), 519-581. · Zbl 1135.11037
[16] Maass, H., Spherical functions and quadratic forms, J. Indian Math. Soc. (N.S.), 20, 117-162, (1956) · Zbl 0072.08401
[17] Maass, H., Über die Verteilung der zweidimensionalen Untergitter in einem euklidischen Gitter, Math. Ann., 137, 319-327, (1959) · Zbl 0085.06902
[18] Marklof, J., The asymptotic distribution of Frobenius numbers, Invent. Math., 181, 1, 179-207, (2010) · Zbl 1200.11022
[19] Margulis, G. A., Discrete Subgroups of Semisimple Lie Groups, (1991), Springer: Springer, Berlin · Zbl 0732.22008
[20] Mozes, S. and Shah, N.. On the space of ergodic invariant measures of unipotent flows. Ergod. Th. & Dynam. Sys.15(1) (1995), 149-159. · Zbl 0818.58028
[21] Margulis, G. A. and Tomanov, G. M.. Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math.116(1-3) (1994), 347-392. · Zbl 0816.22004
[22] Oh, H., Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., 113, 1, 133-192, (2002) · Zbl 1011.22007
[23] Platonov, V. and Rapinchuk, A.. Algebraic Groups and Number Theory. Academic Press, Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. · Zbl 0806.11002
[24] Prasad, G.. Volumes of S-arithmetic quotients of semi-simple groups. Publ. Math. Inst. Hautes Études Sci.69 (1989), 91-117. With an appendix by Moshe Jarden and the author. · Zbl 0695.22005
[25] Ratner, M.. On the p-adic and S-arithmetic generalizations of Raghunathan’s conjectures. Lie Groups and Ergodic Theory (Mumbai, 1996). Tata Institute of Fundamental Research, Bombay, 1998, pp. 167-202. · Zbl 0943.22010
[26] Schmidt, W. M., The distribution of sublattices of Zm, Monatsh. Math., 125, 1, 37-81, (1998) · Zbl 0913.11028
[27] Serre, J. P.. A Course in Arithmetic. Springer, New York, 1973, viii+115 pp. · Zbl 0256.12001
[28] Venkatesh, A., Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172, 2, 989-1094, (2010) · Zbl 1214.11051
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