## Duke Mathematical Journal

### An effective Ratner equidistribution result for $\operatorname{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2}$

Andreas Strömbergsson

#### Abstract

Let $G=\operatorname{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2}$ be the affine special linear group of the plane, and set $\Gamma=\operatorname{SL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$. We prove a polynomially effective asymptotic equidistribution result for the orbits of a $1$-dimensional, nonhorospherical unipotent flow on $\Gamma \backslash G$.

#### Article information

Source
Duke Math. J., Volume 164, Number 5 (2015), 843-902.

Dates
First available in Project Euclid: 7 April 2015

https://projecteuclid.org/euclid.dmj/1428419274

Digital Object Identifier
doi:10.1215/00127094-2885873

Mathematical Reviews number (MathSciNet)
MR3332893

Zentralblatt MATH identifier
1351.37014

#### Citation

Strömbergsson, Andreas. An effective Ratner equidistribution result for $\operatorname{SL}(2,\mathbb{R})\ltimes\mathbb{R}^{2}$. Duke Math. J. 164 (2015), no. 5, 843--902. doi:10.1215/00127094-2885873. https://projecteuclid.org/euclid.dmj/1428419274

#### References

• [1] T. Browning and I. Vinogradov, Effective Ratner theorem for $\mathrm{ASL}(2,\mathbb{R})$ and gaps in $\sqrt{n}$ modulo $1$, preprint, arXiv:1311.6387v2 [math.DS].
• [2] M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J. 61 (1990), 779–803.
• [3] S. G. Dani, On uniformly distributed orbits of certain horocycle flows, Ergodic Theory Dynam. Systems 2 (1982), 139–158.
• [4] S. Edwards, The rate of mixing for diagonal flows on spaces of affine lattices, preprint, http://www.math.uu.se/Research/Publications/Student+theses/ (accessed 11 February 2015).
• [5] M. Einsiedler, G. Margulis, and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math. 177 (2009), 137–212.
• [6] D. El-Baz, J. Marklof, and I. Vinogradov, The distribution of directions in an affine lattice: Two-point correlations and mixed moments, to appear in Int. Math. Res. Not. IMRN, preprint, arXiv:1306.0028v2 [math.NT].
• [7] D. El-Baz, J. Marklof, and I. Vinogradov, The two-point correlation function of the fractional parts of $\sqrt{n}$ is Poisson, to appear in Proc. Amer. Math. Soc., preprint, arXiv:1306.6543v1 [math.NT].
• [8] N. D. Elkies and C. T. McMullen, Gaps in $\sqrt{n}\bmod1$ and ergodic theory, Duke Math. J. 123 (2004), 95–139. Correction, Duke Math. J. 129 (2005), 405–406.
• [9] A. Eskin, “Unipotent flows and applications” in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc. 10, Amer. Math. Soc., Providence, 2010, 71–129.
• [10] A. Eskin and H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems 26 (2006), 163–167.
• [11] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J. 119 (2003), 465–526.
• [12] V. F. Gaposhkin, The dependence of the rate of convergence in the strong law of large numbers for stationary processes on the rate of diminution of the correlation function (in Russian), Teor. Veroyatn. Primen. 26, no. 4 (1981), 720–733; English translation in Theory Probab. Appl. 26, no. 4 (1982), 706–720.
• [13] A. Gorodnik and H. Oh, Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary, Duke Math. J. 139 (2007), 483–525.
• [14] B. Green and T. Tao, Linear equations in primes, Ann. of Math. (2) 171 (2010), 1753–1850.
• [15] B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), 465–540.
• [16] B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), 541–566.
• [17] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, Oxford, 2008.
• [18] D. A. Hejhal, On the uniform equidistribution of long closed horocycles, Asian J. Math. 4 (2000), 839–853.
• [19] R. Howe and E. Tan, Nonabelian Harmonic Analysis, Universitext, Springer, New York, 1992.
• [20] A. E. Ingham, The Distribution of Prime Numbers, reprint of the 1932 original, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1990.
• [21] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, Amer. Math. Soc., Providence, 2004.
• [22] V. Jarnik, Über die simultanen diophantischen Approximationen, Math. Z. 33 (1931), 505–543.
• [23] D. Kleinbock, N. Shah, and A. Starkov, “Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory” in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 813–930.
• [24] A. W. Knapp, Lie Groups beyond an Introduction, 2nd ed., Progr. Math. 140, Birkhäuser, Boston, 2002.
• [25] E. Lindenstrauss and G. Margulis, Effective estimates on indefinite ternary forms, Israel J. Math. 203 (2014), 445–499.
• [26] G. Margulis, “Problems and conjectures in rigidity theory” in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, 2000, 161–174.
• [27] G. Margulis, On Some Aspects of the Theory of Anosov Systems, with a survey by R. Sharp, Springer Monogr. Math., Springer, Berlin, 2004.
• [28] G. Margulis and A. Mohammadi, Quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms, Duke Math. J. 158 (2011), 121–160.
• [29] J. Marklof, Pair correlation densities of inhomogeneous quadratic forms, II, Duke Math. J. 115 (2002), 409–434.
• [30] J. Marklof, Pair correlation densities of inhomogeneous quadratic forms, Ann. of Math. (2) 158 (2003), 419–471.
• [31] J. Marklof, Mean square value of exponential sums related to the representation of integers as sums of squares, Acta Arith. 117 (2005), 353–370.
• [32] J. Marklof and A. Strömbergsson, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems, Ann. of Math. (2) 172 (2010), 1949–2033.
• [33] J. Marklof and A. Strömbergsson, Free path lengths in quasicrystals, Comm. Math. Phys. 330 (2014), 723–755.
• [34] A. Mohammadi, A special case of effective equidistribution with explicit constants, Ergodic Theory Dynam. Systems 32 (2012), 237–247.
• [35] D. W. Morris, Ratner’s Theorems on Unipotent Flows, Chicago Lectures in Math., University of Chicago Press, Chicago, 2005.
• [36] M. B. Nathanson, Additive Number Theory, Grad. Texts in Math. 164, Springer, New York, 1996.
• [37] M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb. (3) 68, Springer, New York, 1972.
• [38] M. Ratner, The rate of mixing for geodesic and horocycle flow, Ergodic Theory Dynam. Systems 7 (1987), 267–288.
• [39] M. Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), 545–607.
• [40] M. Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), 235–280.
• [41] P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math. 34 (1981), 719–739.
• [42] P. Sarnak and A. Ubis, The horocycle flow at prime times, J. Math. Pures Appl. (9) 103 (2015), 575–618.
• [43] N. Shah, Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 105–125.
• [44] N. Shah, Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds, Duke Math. J. 148 (2009), 281–304.
• [45] N. Shah, Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms, J. Amer. Math. Soc. 23 (2010), 563–589.
• [46] Y. G. Sinai, “Statistics of gaps in the sequence $\{\sqrt{n}\}$” in Dynamical Systems and Group Actions, Contemp. Math. 567, Amer. Math. Soc., Providence, 2012, 185–189.
• [47] A. Strömbergsson, On the uniform equidistribution of long closed horocycles, Duke Math. J. 123 (2004), 507–547.
• [48] A. Strömbergsson, On the deviation of ergodic averages for horocycle flows, J. Mod. Dyn. 7 (2013), 291–328.
• [49] A. Strömbergsson and A. Venkatesh, Small solutions to linear congruences and Hecke equidistribution, Acta Arith., 118 (2005), 41–78.
• [50] A. Strömbergsson and P. Vishe, Effective Ratner equidistribution for $\operatorname{SL}(2,\mathbb{R})\ltimes(\mathbb{R} ^{2})^{\oplus k}$ and applications to quadratic forms, in preparation.
• [51] A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2) 172 (2010), 989–1094.
• [52] A. Weil, On some exponential sums, Proc. Natl. Acad. Sci. USA 34 (1948), 204–207.