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On Omri Sarig's work on the dynamics on surfaces
Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows
1. | School of Mathematics, University of Bristol, Bristol BS8 1TW |
2. | Département de mathématique, UMR 8628 CNRS, Bât. 425, Université Paris-Sud, 91405 ORSAY Cedex, France |
References:
[1] |
T. Apostol, Introduction to Analytic Number Theory,, Undergrad. Texts Math., (1976).
|
[2] |
M. Babillot, Points entiers et groupes discrets: De l'analyse aux systèmes dynamiques,, in Rigidité, (2002), 1.
|
[3] |
B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T),, New Math. Mono., (2008).
doi: 10.1017/CBO9780511542749. |
[4] |
Y. Benoist and H. Oh, Effective equidistribution of $S$-integral points on symmetric varieties,, Ann. Inst. Fourier (Grenoble), 62 (2012), 1889.
doi: 10.5802/aif.2738. |
[5] |
A. Borel, Ensembles fundamentaux pour les groupes arithmétiques,, in Colloque sur la Théorie des Groupes Algébriques (Bruxelles, (1962), 23.
|
[6] |
A. Borel, Introduction aux Groupes Arithmétiques,, Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1341).
|
[7] |
A. Borel, Linear Algebraic Groups,, 2nd edition, (1991).
doi: 10.1007/978-1-4612-0941-6. |
[8] |
A. Borel, Reduction theory for arithmetic groups,, in Algebraic Groups and Discontinuous Subgroups (eds. A. Borel and G. D. Mostow) (Proc. Sympos. Pure Math. Boulder, (1965), 20.
|
[9] |
A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups,, Ann. of Math. (2), 75 (1962), 485.
doi: 10.2307/1970210. |
[10] |
A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces,, Mathematics: Theory & Applications, (2006).
|
[11] |
M. Borovoi and Z. Rudnick, Hardy-Littlewood varieties and semisimple groups,, Invent. Math., 119 (1995), 37.
doi: 10.1007/BF01245174. |
[12] |
L. Clozel, Démonstration de la conjecture $\tau$,, Invent. Math., 151 (2003), 297.
doi: 10.1007/s00222-002-0253-8. |
[13] |
H. Cohn, A Second Course in Number Theory,, Wiley, (1962).
|
[14] |
J.-L. Colliot-Thélène and F. Xu, Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms,, Compositio Math., 145 (2009), 309.
doi: 10.1112/S0010437X0800376X. |
[15] |
M. Cowling, Sur les coefficients des représentations unitaires des groupes de Lie simples,, in Analyse Harmonique sur les Groupes de Lie (Sém. Nancy-Strasbourg 1976-1978), (1979), 1976.
|
[16] |
W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties,, Duke Math. J., 71 (1993), 143.
doi: 10.1215/S0012-7094-93-07107-4. |
[17] |
A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups,, Duke Math. J., 71 (1993), 181.
doi: 10.1215/S0012-7094-93-07108-6. |
[18] |
A. Eskin, S. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties,, Ann. of Math. (2), 143 (1996), 253.
doi: 10.2307/2118644. |
[19] |
A. Eskin and H. Oh, Representations of integers by an invariant polynomial and unipotent flows,, Duke Math. J., 135 (2006), 481.
doi: 10.1215/S0012-7094-06-13533-0. |
[20] |
A. Eskin, Z. Rudnick and P. Sarnak, A proof of Siegel's weight formula,, Internat. Math. Res. Notices, 5 (1991), 65.
doi: 10.1155/S1073792891000090. |
[21] |
W. T. Gan and H. Oh, Equidistribution of integer points on a family of homogeneous varieties: A problem of Linnik,, Compositio Math., 136 (2003), 323.
doi: 10.1023/A:1023256605535. |
[22] |
A. Gorodnik and H. Oh, Rational points on homogeneous varieties and equidistribution of adelic periods,, Geom. Funct. Anal., 21 (2011), 319.
doi: 10.1007/s00039-011-0113-z. |
[23] |
K. Györy, On the distribution of solutions of decomposable form equations,, in Number Theory in Progress, (1997), 237.
|
[24] |
M. Hirsch, Differential Topology,, Grad. Texts Math., (1976).
|
[25] |
D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of $ PSL(2,\RR)$,, J. Euro. Math. Soc., 11 (2009), 283.
doi: 10.4171/JEMS/151. |
[26] |
T. Kimura, Introduction to Prehomogeneous Vector Spaces,, Transl. Math. Mono., (2003).
|
[27] |
D. Kleinbock and G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in Sinaĭ's Moscow Seminar on Dynamical Systems, (1996), 141.
|
[28] |
D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces,, Invent. Math., 138 (1999), 451.
doi: 10.1007/s002220050350. |
[29] |
H. Koch, Number Theory: Algebraic Numbers and Functions,, Grad. Stud. Math., (2000).
|
[30] |
S. Lang, Algebraic Number Theory,, Second edition, (1994).
doi: 10.1007/978-1-4612-0853-2. |
[31] |
D. N. Lehmer, Asymptotic evaluation of certain totient sums,, Amer. J. Math., 22 (1900), 293.
doi: 10.2307/2369728. |
[32] |
A. Nevo, Exponential volume growth, maximal functions on symmetric spaces, and ergodic theorems for semi-simple Lie groups,, Erg. Theo. Dyn. Syst., 25 (2005), 1257.
doi: 10.1017/S0143385704000951. |
[33] |
H. Oh, Hardy-Littlewood system and representations of integers by an invariant polynomial,, Geom. Funct. Anal., 14 (2004), 791.
doi: 10.1007/s00039-004-0475-6. |
[34] |
H. Oh, Orbital counting via mixing and unipotent flows,, in Homogeneous Flows, (2010), 339.
|
[35] |
E. Peyre, Obstructions au principe de Hasse et à l'approximation faible,, Séminaire Bourbaki, 299 (2005), 165.
|
[36] |
J. Parkkonen and F. Paulin, Équidistribution, comptage et approximation par irrationnels quadratiques,, J. Mod. Dyn., 6 (2012), 1.
doi: 10.3934/jmd.2012.6.1. |
[37] |
J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature,, preprint, (). Google Scholar |
[38] |
J. Parkkonen and F. Paulin, On the arithmetic of crossratios and generalised Mertens' formulas,, to appear in Ann. Fac. Scien. Toulouse, (2013). Google Scholar |
[39] |
V. Platonov and A. Rapinchuck, Algebraic Groups and Number Theory,, Pure and Applied Mathematics, (1994).
|
[40] |
M. Raghunathan, Discrete Subgroups of Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1972).
|
[41] |
I. Reiner, Maximal Orders,, Academic Press, (1975).
|
[42] |
P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series,, Comm. Pure Appl. Math., 34 (1981), 719.
doi: 10.1002/cpa.3160340602. |
[43] |
M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces,, Ann. of Math. (2), 100 (1974), 131.
doi: 10.2307/1970844. |
[44] |
W. M. Schmidt, Norm form equation,, Ann. of Math. (2), 96 (1972), 526.
doi: 10.2307/1970824. |
[45] |
J.-P. Serre, Cours d'arithmetique,, Collection SUP:, (1970).
|
[46] |
C. L. Siegel, On the theory of indefinite quadratic forms,, Ann. of Math. (2), 45 (1944), 577.
doi: 10.2307/1969191. |
[47] |
C. L. Siegel, The average measure of quadratic forms with given determinant and signature,, Ann. of Math. (2), 45 (1944), 667.
doi: 10.2307/1969296. |
[48] |
T. A. Springer, Linear algebraic groups,, in Algebraic Geometry IV (eds. A. Parshin and I. Shavarevich), (1994), 1.
doi: 10.1007/978-3-662-03073-8. |
[49] |
J. L. Thunder, Decomposable form inequalities,, Ann. of Math. (2), 153 (2001), 767.
doi: 10.2307/2661368. |
[50] |
V. E. Voskresenskiĭ, Algebraic Groups and their Birational Invariants,, Transl. Math. Mono., (1998).
|
[51] |
A. Weil, L'intégration dans les groupes topologiques et ses applications,, Hermann, (1965). Google Scholar |
show all references
References:
[1] |
T. Apostol, Introduction to Analytic Number Theory,, Undergrad. Texts Math., (1976).
|
[2] |
M. Babillot, Points entiers et groupes discrets: De l'analyse aux systèmes dynamiques,, in Rigidité, (2002), 1.
|
[3] |
B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T),, New Math. Mono., (2008).
doi: 10.1017/CBO9780511542749. |
[4] |
Y. Benoist and H. Oh, Effective equidistribution of $S$-integral points on symmetric varieties,, Ann. Inst. Fourier (Grenoble), 62 (2012), 1889.
doi: 10.5802/aif.2738. |
[5] |
A. Borel, Ensembles fundamentaux pour les groupes arithmétiques,, in Colloque sur la Théorie des Groupes Algébriques (Bruxelles, (1962), 23.
|
[6] |
A. Borel, Introduction aux Groupes Arithmétiques,, Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1341).
|
[7] |
A. Borel, Linear Algebraic Groups,, 2nd edition, (1991).
doi: 10.1007/978-1-4612-0941-6. |
[8] |
A. Borel, Reduction theory for arithmetic groups,, in Algebraic Groups and Discontinuous Subgroups (eds. A. Borel and G. D. Mostow) (Proc. Sympos. Pure Math. Boulder, (1965), 20.
|
[9] |
A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups,, Ann. of Math. (2), 75 (1962), 485.
doi: 10.2307/1970210. |
[10] |
A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces,, Mathematics: Theory & Applications, (2006).
|
[11] |
M. Borovoi and Z. Rudnick, Hardy-Littlewood varieties and semisimple groups,, Invent. Math., 119 (1995), 37.
doi: 10.1007/BF01245174. |
[12] |
L. Clozel, Démonstration de la conjecture $\tau$,, Invent. Math., 151 (2003), 297.
doi: 10.1007/s00222-002-0253-8. |
[13] |
H. Cohn, A Second Course in Number Theory,, Wiley, (1962).
|
[14] |
J.-L. Colliot-Thélène and F. Xu, Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms,, Compositio Math., 145 (2009), 309.
doi: 10.1112/S0010437X0800376X. |
[15] |
M. Cowling, Sur les coefficients des représentations unitaires des groupes de Lie simples,, in Analyse Harmonique sur les Groupes de Lie (Sém. Nancy-Strasbourg 1976-1978), (1979), 1976.
|
[16] |
W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties,, Duke Math. J., 71 (1993), 143.
doi: 10.1215/S0012-7094-93-07107-4. |
[17] |
A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups,, Duke Math. J., 71 (1993), 181.
doi: 10.1215/S0012-7094-93-07108-6. |
[18] |
A. Eskin, S. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties,, Ann. of Math. (2), 143 (1996), 253.
doi: 10.2307/2118644. |
[19] |
A. Eskin and H. Oh, Representations of integers by an invariant polynomial and unipotent flows,, Duke Math. J., 135 (2006), 481.
doi: 10.1215/S0012-7094-06-13533-0. |
[20] |
A. Eskin, Z. Rudnick and P. Sarnak, A proof of Siegel's weight formula,, Internat. Math. Res. Notices, 5 (1991), 65.
doi: 10.1155/S1073792891000090. |
[21] |
W. T. Gan and H. Oh, Equidistribution of integer points on a family of homogeneous varieties: A problem of Linnik,, Compositio Math., 136 (2003), 323.
doi: 10.1023/A:1023256605535. |
[22] |
A. Gorodnik and H. Oh, Rational points on homogeneous varieties and equidistribution of adelic periods,, Geom. Funct. Anal., 21 (2011), 319.
doi: 10.1007/s00039-011-0113-z. |
[23] |
K. Györy, On the distribution of solutions of decomposable form equations,, in Number Theory in Progress, (1997), 237.
|
[24] |
M. Hirsch, Differential Topology,, Grad. Texts Math., (1976).
|
[25] |
D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of $ PSL(2,\RR)$,, J. Euro. Math. Soc., 11 (2009), 283.
doi: 10.4171/JEMS/151. |
[26] |
T. Kimura, Introduction to Prehomogeneous Vector Spaces,, Transl. Math. Mono., (2003).
|
[27] |
D. Kleinbock and G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in Sinaĭ's Moscow Seminar on Dynamical Systems, (1996), 141.
|
[28] |
D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces,, Invent. Math., 138 (1999), 451.
doi: 10.1007/s002220050350. |
[29] |
H. Koch, Number Theory: Algebraic Numbers and Functions,, Grad. Stud. Math., (2000).
|
[30] |
S. Lang, Algebraic Number Theory,, Second edition, (1994).
doi: 10.1007/978-1-4612-0853-2. |
[31] |
D. N. Lehmer, Asymptotic evaluation of certain totient sums,, Amer. J. Math., 22 (1900), 293.
doi: 10.2307/2369728. |
[32] |
A. Nevo, Exponential volume growth, maximal functions on symmetric spaces, and ergodic theorems for semi-simple Lie groups,, Erg. Theo. Dyn. Syst., 25 (2005), 1257.
doi: 10.1017/S0143385704000951. |
[33] |
H. Oh, Hardy-Littlewood system and representations of integers by an invariant polynomial,, Geom. Funct. Anal., 14 (2004), 791.
doi: 10.1007/s00039-004-0475-6. |
[34] |
H. Oh, Orbital counting via mixing and unipotent flows,, in Homogeneous Flows, (2010), 339.
|
[35] |
E. Peyre, Obstructions au principe de Hasse et à l'approximation faible,, Séminaire Bourbaki, 299 (2005), 165.
|
[36] |
J. Parkkonen and F. Paulin, Équidistribution, comptage et approximation par irrationnels quadratiques,, J. Mod. Dyn., 6 (2012), 1.
doi: 10.3934/jmd.2012.6.1. |
[37] |
J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature,, preprint, (). Google Scholar |
[38] |
J. Parkkonen and F. Paulin, On the arithmetic of crossratios and generalised Mertens' formulas,, to appear in Ann. Fac. Scien. Toulouse, (2013). Google Scholar |
[39] |
V. Platonov and A. Rapinchuck, Algebraic Groups and Number Theory,, Pure and Applied Mathematics, (1994).
|
[40] |
M. Raghunathan, Discrete Subgroups of Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1972).
|
[41] |
I. Reiner, Maximal Orders,, Academic Press, (1975).
|
[42] |
P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series,, Comm. Pure Appl. Math., 34 (1981), 719.
doi: 10.1002/cpa.3160340602. |
[43] |
M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces,, Ann. of Math. (2), 100 (1974), 131.
doi: 10.2307/1970844. |
[44] |
W. M. Schmidt, Norm form equation,, Ann. of Math. (2), 96 (1972), 526.
doi: 10.2307/1970824. |
[45] |
J.-P. Serre, Cours d'arithmetique,, Collection SUP:, (1970).
|
[46] |
C. L. Siegel, On the theory of indefinite quadratic forms,, Ann. of Math. (2), 45 (1944), 577.
doi: 10.2307/1969191. |
[47] |
C. L. Siegel, The average measure of quadratic forms with given determinant and signature,, Ann. of Math. (2), 45 (1944), 667.
doi: 10.2307/1969296. |
[48] |
T. A. Springer, Linear algebraic groups,, in Algebraic Geometry IV (eds. A. Parshin and I. Shavarevich), (1994), 1.
doi: 10.1007/978-3-662-03073-8. |
[49] |
J. L. Thunder, Decomposable form inequalities,, Ann. of Math. (2), 153 (2001), 767.
doi: 10.2307/2661368. |
[50] |
V. E. Voskresenskiĭ, Algebraic Groups and their Birational Invariants,, Transl. Math. Mono., (1998).
|
[51] |
A. Weil, L'intégration dans les groupes topologiques et ses applications,, Hermann, (1965). Google Scholar |
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