Publications Mathématiques de Besançon

Algèbre et Théorie des Nombres
p-adic Directions of Primitive Vectors
[Directions p-adique de vecteurs primitifs]
Antonin Guilloux1 ; Tal Horesh2
1 IMJ-PRG, OURAGAN, Sorbonne Université, CNRS, INRIA, France
2 IST Austria
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2023), pp. 85-107.

Les problèmes de type Linnik concernent la distribution des projections des points entiers sur la sphère unitaire lorsque leur norme augmente et différentes généralisations de ce phénomène. Notre travail s’intéresse à une question de ce type : nous prouvons la distribution uniforme des projections des points primitifs de ℤ 2 sur la sphère unitaire p-adique lorsque leur norme (réelle) tend vers l’infini. La preuve se fait en comptant les points d’un réseau dans des S-groupes arithmétiques semi-simples.

Linnik type problems concern the distribution of projections of integral points on the unit sphere as their norm increases, and different generalizations of this phenomenon. Our work addresses a question of this type: we prove the uniform distribution of the projections of primitive ℤ 2 points in the p-adic unit sphere, as their (real) norm tends to infinity. The proof is via counting lattice points in semi-simple S-arithmetic groups.

Publié le :
DOI : 10.5802/pmb.50
Antonin Guilloux 1 ; Tal Horesh 2

1 IMJ-PRG, OURAGAN, Sorbonne Université, CNRS, INRIA, France
2 IST Austria
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Antonin Guilloux; Tal Horesh. $p$-adic Directions of Primitive Vectors. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2023), pp. 85-107. doi : 10.5802/pmb.50. https://pmb.centre-mersenne.org/articles/10.5802/pmb.50/

[1] Menny Aka; Manfred Einsiedler; Uri Shapira Integer points on spheres and their orthogonal grids, J. Lond. Math. Soc., Volume 93 (2016) no. 2, pp. 143-158 | MR | Zbl

[2] Menny Aka; Manfred Einsiedler; Uri Shapira Integer points on spheres and their orthogonal lattices, Invent. Math., Volume 206 (2016) no. 2, pp. 379-396 | MR | Zbl

[3] Yves Benoist; Hee Oh Effective equidistribution of S-integral points on symmetric varieties, Ann. Inst. Fourier, Volume 62 (2012) no. 5, pp. 1889-1942 | DOI | Numdam | MR | Zbl

[4] Armand Borel; Harish-Chandra Arithmetic subgroups of algebraic groups, Ann. Math., Volume 75 (1962), pp. 485-535 | DOI | MR | Zbl

[5] John Bryk; Cesar E. Silva Measurable dynamics of simple p-adic polynomials, Am. Math. Mon., Volume 112 (2005) no. 3, pp. 212-232 | DOI | MR | Zbl

[6] Laurent Clozel; Hee Oh; Emmanuel Ullmo Hecke operators and equidistribution of Hecke points, Invent. Math., Volume 144 (2001) no. 2, pp. 327-351 | DOI | MR | Zbl

[7] William Duke Rational points on the sphere, Ramanujan J., Volume 7 (2003) no. 1, pp. 235-239 | DOI | MR | Zbl

[8] William Duke An introduction to the Linnik problems, Equidistribution in number theory, an introduction (NATO Science Series II: Mathematics, Physics and Chemistry), Volume 237, Springer, 2007, pp. 197-216 | DOI | MR | Zbl

[9] Manfred Einsiedler; Elon Lindenstrauss; Philippe Michel; Akshay Venkatesh Distribution of periodic torus orbits on homogeneous spaces, Duke Math. J., Volume 148 (2009) no. 1, pp. 119-174 | MR | Zbl

[10] Manfred Einsiedler; Shahar Mozes; Nimish Shah; Uri Shapira Equidistribution of primitive rational points on expanding horospheres, Compos. Math., Volume 152 (2016) no. 4, pp. 667-692 | DOI | MR | Zbl

[11] Manfred Einsiedler; René Rühr; Philipp Wirth Distribution of shapes of orthogonal lattices, Ergodic Theory Dyn. Syst., Volume 39 (2019) no. 6, pp. 1531-1607 | DOI | MR | Zbl

[12] Manfred Einsiedler; Thomas Ward Ergodic theory (with a view towards number theory), Graduate Texts in Mathematics, 259, Springer, 2011 | DOI

[13] Alex Eskin; Curt McMullen Mixing, Counting and Equidistribution in Lie Groups, Duke Math. J., Volume 71 (1993) no. 1, pp. 181-209 | MR | Zbl

[14] Stephen Gelbart; Hervé Jacquet A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Éc. Norm. Supér., Volume 11 (1978) no. 4, pp. 471-542 | DOI | Numdam | MR | Zbl

[15] Oscar Goldman; Nagayoshi Iwahori The space of 𝔭-adic norms, Acta Math., Volume 109 (1963) no. 1, pp. 137-177 | DOI | MR

[16] Anton Good On various means involving the Fourier coefficients of cusp forms, Math. Z., Volume 183 (1983) no. 1, pp. 95-129 | DOI | MR | Zbl

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

[18] Antonin Guilloux Equidistribution in S-arithmetic and adelic spaces, Ann. Fac. Sci. Toulouse, Math., Volume 23 (2014) no. 5, pp. 1023-1048 | DOI | Numdam | MR | Zbl

[19] Tal Horesh; Yakov Karasik Horospherical coordinates of lattice points in hyperbolic space: effective counting and equidistribution (2016) (http://arxiv.org/abs/1612.08215)

[20] Tal Horesh; Yakov Karasik Equidistribution of primitive vectors, and the shortest solutions to their GCD equations (2019) (http://arxiv.org/abs/1903.01560)

[21] Tal Horesh; Yakov Karasik A practical guide to well roundedness (2020) (http://arxiv.org/abs/2011.12204)

[22] Henry H. Kim Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2 , J. Am. Math. Soc., Volume 16 (2003) no. 1, pp. 139-183 (with an appendix by D. Ramakrishnan, and an appendix co-authored by P. Sarnak) | MR

[23] Yuriĭ V. Linnik Ergodic Properties of Algebraic Fields, Ergebnisse der Mathematik und ihrer Grenzgebiete, 45, Springer, 1968

[24] Jens Marklof The asymptotic distribution of Frobenius numbers, Invent. Math., Volume 181 (2010) no. 1, pp. 179-207 | DOI | MR | Zbl

[25] Jens Marklof; Ilya Vinogradov Directions in Hyperbolic Lattices, J. Reine Angew. Math., Volume 740 (2018), pp. 161-186 | DOI | MR | Zbl

[26] Vladimir P. Platonov; Andrei S. Rapinchuk Algebraic groups and number theory, Russ. Math. Surv., Volume 47 (1992) no. 2, pp. 133-161 | DOI | Zbl

[27] Morten S. Risager; Zeév Rudnick On the statistics of the minimal solution of a linear Diophantine equation and uniform distribution of the real part of orbits in hyperbolic spaces, Spectral analysis in geometry and number theory (Contemporary Mathematics), Volume 484, American Mathematical Society, 2009, pp. 187-194 | DOI | MR

[28] Wolfgang M. Schmidt The distribution of sub-lattices of Z m , Monatsh. Math., Volume 125 (1998) no. 1, pp. 37-81 | DOI

[29] Jimi L. Truelsen Effective equidistribution of the real part of orbits on hyperbolic surfaces, Proc. Am. Math. Soc., Volume 141 (2013) no. 2, pp. 505-514 | DOI | MR | Zbl

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