2015, 9: 305-353. doi: 10.3934/jmd.2015.9.305

Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds

1. 

Centro de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal

2. 

UMR CNRS 8524, UFR de Mathématiques, Université de Lille 1, F59655 Villeneuve d’Asq CEDEX, France

Received  June 2015 Revised  September 2015 Published  November 2015

We prove quantitative equidistribution results for actions of Abelian subgroups of the $(2g+1)$-dimensional Heisenberg group acting on compact $(2g+1)$-dimensional homogeneous nilmanifolds. The results are based on the study of the $C^\infty$-cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalization method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in $g$ variables, generalizing the classical results of Hardy and Littlewood [25,26] and the optimal result of Fiedler, Jurkat, and Körner [17] to higher dimension.
Citation: Salvatore Cosentino, Livio Flaminio. Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds. Journal of Modern Dynamics, 2015, 9: 305-353. doi: 10.3934/jmd.2015.9.305
References:
[1]

L. Auslander and R. Tolimieri, Abelian harmonic analysis, theta functions and function algebras on a nilmanifold,, Lecture Notes in Mathematics, (1975).

[2]

L. Auslander, Lecture Notes on Nil-Theta Functions,, Regional Conference Series in Mathematics, (1977).

[3]

A. Bufetov and G. Forni, Théorèmes limites pour les flots horocycliques,, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 851.

[4]

M. V. Berry and J. Goldberg, Renormalisation of curlicues,, Nonlinearity, 1 (1988), 1. doi: 10.1088/0951-7715/1/1/001.

[5]

H. Cartan, Ouverts fondamentaux pour le groupe modulaire,, Séminaire Henri Cartan, 10 (): 1957.

[6]

N. Chevallier, Meilleures approximations diophantiennes simultanées et théorème de Lévy,, Ann. Inst. Fourier (Grenoble), 55 (2005), 1635. doi: 10.5802/aif.2134.

[7]

N. Chevallier, Best simultaneous Diophantine approximations and multidimensional continued fraction expansions,, Mosc. J. Comb. Number Theory, 3 (2013), 3.

[8]

E. A. Coutsias and N. D. Kazarinoff, The approximate functional formula for the theta function and Diophantine Gauss sums,, Trans. Amer. Math. Soc., 350 (1998), 615. doi: 10.1090/S0002-9947-98-02024-8.

[9]

F. Cellarosi and J. Marklof, Quadratic Weyl sums, automorphic functions, and invariance principles,, , (2015).

[10]

S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation,, J. Reine Angew. Math., 359 (1985), 55. doi: 10.1515/crll.1985.359.55.

[11]

D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions I. KAM method and $\mathbb Z^k$ actions on the torus,, Ann. of Math. (2), 172 (2010), 1805. doi: 10.4007/annals.2010.172.1805.

[12]

_______, Local rigidity of homogeneous parabolic actions: I. A model case,, J. Mod. Dyn., 5 (2011), 203. doi: 10.3934/jmd.2011.5.203.

[13]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. doi: 10.1215/S0012-7094-03-11932-8.

[14]

________, Equidistribution of nilflows and applications to theta sums,, Ergodic Theory Dynam. Systems, 26 (2006), 409. doi: 10.1017/S014338570500060X.

[15]

________, On the cohomological equation for nilflows,, J. Mod. Dyn., 1 (2007), 37.

[16]

________, On effective equidistribution for higher step nilflows,, , (2014).

[17]

H. Fiedler, W. Jurkat and O. Körner, Asymptotic expansions of finite theta series,, Acta Arith., 32 (1977), 129.

[18]

A. Fedotov and F. Klopp, An exact renormalization formula for Gaussian exponential sums and applications,, Amer. J. Math., 134 (2012), 711. doi: 10.1353/ajm.2012.0016.

[19]

G. B. Folland, Harmonic analysis in phase space,, Annals of Mathematics Studies, (1989).

[20]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150.

[21]

F. Götze and M. Gordin, Limiting distributions of theta series on Siegel half-spaces,, Algebra i Analiz, 15 (2003), 118. doi: 10.1090/S1061-0022-03-00803-3.

[22]

F. Götze and G. Margulis, Distribution of values of quadratic forms at integral points,, , (2010).

[23]

J. Griffin and J. Marklof, Limit theorems for skew translations,, J. Mod. Dyn. 8 (2014), 8 (2014), 177. doi: 10.3934/jmd.2014.8.177.

[24]

R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65. doi: 10.1090/S0273-0979-1982-15004-2.

[25]

G. H. Hardy and J. E. Littlewood, Some problems of diophantine approximation,, Acta Math., 37 (1914), 193. doi: 10.1007/BF02401834.

[26]

________, Some problems of diophantine approximation: An additional note on the trigonometrical series associated with the elliptic theta-functions,, Acta Math., 47 (1926), 189. doi: 10.1007/BF02544111.

[27]

A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in collaboration with E. A. Robinson, (1999), 107. doi: 10.1090/pspum/069/1858535.

[28]

________, Combinatorial Constructions in Ergodic Theory and Dynamics,, University Lecture Series, (2003). doi: 10.1090/ulect/030.

[29]

A. Katok and S. Katok, Higher cohomology for abelian groups of toral automorphisms,, Ergodic Theory Dynam. Systems, 15 (1995), 569. doi: 10.1017/S0143385700008531.

[30]

________, Higher cohomology for abelian groups of toral automorphisms. II. The partially hyperbolic case, and corrigendum,, Ergodic Theory Dynam. Systems, 25 (2005), 1909. doi: 10.1017/S0143385705000271.

[31]

H. Klingen, Introductory Lectures on Siegel Modular Forms,, Cambridge Studies in Advanced Mathematics, (1990). doi: 10.1017/CBO9780511619878.

[32]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces,, Invent. Math., 138 (1999), 451. doi: 10.1007/s002220050350.

[33]

A. Katok and V. Niţică, Rigidity in Higher Rank Abelian Group Actions. Volume I. Introduction and Cocycle Problem,, Cambridge Tracts in Mathematics, (2011). doi: 10.1017/CBO9780511803550.

[34]

A. Katok and F. Rodriguez Hertz, Rigidity of real-analytic actions of $SL(n,\mathbb Z)$ on $\mathbb T^n$: A case of realization of Zimmer program,, Discrete Contin. Dyn. Syst., 27 (2010), 609. doi: 10.3934/dcds.2010.27.609.

[35]

A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions,, Tr. Mat. Inst. Steklova, 216 (1997), 292.

[36]

J. C. Lagarias, Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators,, Trans. Amer. Math. Soc., 272 (1982), 545. doi: 10.2307/1998713.

[37]

G. W. Mackey, A theorem of Stone and von Neumann,, Duke Math. J., 16 (1949), 313. doi: 10.1215/S0012-7094-49-01631-2.

[38]

J. Marklof, Limit theorems for theta sums,, Duke Math. J., 97 (1999), 127. doi: 10.1215/S0012-7094-99-09706-5.

[39]

________, Theta sums, Eisenstein series, and the semiclassical dynamics of a precessing spin,, in Emerging Applications of Number Theory (Minneapolis, (1996), 405. doi: 10.1007/978-1-4612-1544-8_17.

[40]

J. Marklof, Pair correlation densities of inhomogeneous quadratic forms,, Ann. Math. (2), 158 (2003), 419. doi: 10.4007/annals.2003.158.419.

[41]

J. Moser, On commuting circle mappings and simultaneous Diophantine approximations,, Math. Z., 205 (1990), 105. doi: 10.1007/BF02571227.

[42]

D. Mumford, Tata Lectures on Theta. I,, With the collaboration of C. Musili, (1983). doi: 10.1007/978-1-4899-2843-6.

[43]

________, Tata Lectures on Theta. III,, With collaboration of M. Nori and P. Norman, (1991).

[44]

L. Schwartz, Théorie des Distributions,, Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1966).

[45]

N. A. Shah, Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds,, Duke Math. J., 148 (2009), 281. doi: 10.1215/00127094-2009-027.

[46]

C. L. Siegel, Symplectic Geometry,, Academic Press, (1964).

[47]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics,, Acta Math., 149 (1982), 215. doi: 10.1007/BF02392354.

[48]

R. Tolimieri, Heisenberg manifolds and theta functions,, Trans. Amer. Math. Soc., 239 (1978), 293. doi: 10.1090/S0002-9947-1978-0487050-7.

[49]

A. Weil, Sur certains groupes d'opérateurs unitaires,, Acta Math., 111 (1964), 143. doi: 10.1007/BF02391012.

[50]

T. D. Wooley, Perturbations of Weyl sums,, Int. Math. Res. Notices, (2015). doi: 10.1093/imrn/rnv225.

show all references

References:
[1]

L. Auslander and R. Tolimieri, Abelian harmonic analysis, theta functions and function algebras on a nilmanifold,, Lecture Notes in Mathematics, (1975).

[2]

L. Auslander, Lecture Notes on Nil-Theta Functions,, Regional Conference Series in Mathematics, (1977).

[3]

A. Bufetov and G. Forni, Théorèmes limites pour les flots horocycliques,, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 851.

[4]

M. V. Berry and J. Goldberg, Renormalisation of curlicues,, Nonlinearity, 1 (1988), 1. doi: 10.1088/0951-7715/1/1/001.

[5]

H. Cartan, Ouverts fondamentaux pour le groupe modulaire,, Séminaire Henri Cartan, 10 (): 1957.

[6]

N. Chevallier, Meilleures approximations diophantiennes simultanées et théorème de Lévy,, Ann. Inst. Fourier (Grenoble), 55 (2005), 1635. doi: 10.5802/aif.2134.

[7]

N. Chevallier, Best simultaneous Diophantine approximations and multidimensional continued fraction expansions,, Mosc. J. Comb. Number Theory, 3 (2013), 3.

[8]

E. A. Coutsias and N. D. Kazarinoff, The approximate functional formula for the theta function and Diophantine Gauss sums,, Trans. Amer. Math. Soc., 350 (1998), 615. doi: 10.1090/S0002-9947-98-02024-8.

[9]

F. Cellarosi and J. Marklof, Quadratic Weyl sums, automorphic functions, and invariance principles,, , (2015).

[10]

S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation,, J. Reine Angew. Math., 359 (1985), 55. doi: 10.1515/crll.1985.359.55.

[11]

D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions I. KAM method and $\mathbb Z^k$ actions on the torus,, Ann. of Math. (2), 172 (2010), 1805. doi: 10.4007/annals.2010.172.1805.

[12]

_______, Local rigidity of homogeneous parabolic actions: I. A model case,, J. Mod. Dyn., 5 (2011), 203. doi: 10.3934/jmd.2011.5.203.

[13]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. doi: 10.1215/S0012-7094-03-11932-8.

[14]

________, Equidistribution of nilflows and applications to theta sums,, Ergodic Theory Dynam. Systems, 26 (2006), 409. doi: 10.1017/S014338570500060X.

[15]

________, On the cohomological equation for nilflows,, J. Mod. Dyn., 1 (2007), 37.

[16]

________, On effective equidistribution for higher step nilflows,, , (2014).

[17]

H. Fiedler, W. Jurkat and O. Körner, Asymptotic expansions of finite theta series,, Acta Arith., 32 (1977), 129.

[18]

A. Fedotov and F. Klopp, An exact renormalization formula for Gaussian exponential sums and applications,, Amer. J. Math., 134 (2012), 711. doi: 10.1353/ajm.2012.0016.

[19]

G. B. Folland, Harmonic analysis in phase space,, Annals of Mathematics Studies, (1989).

[20]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150.

[21]

F. Götze and M. Gordin, Limiting distributions of theta series on Siegel half-spaces,, Algebra i Analiz, 15 (2003), 118. doi: 10.1090/S1061-0022-03-00803-3.

[22]

F. Götze and G. Margulis, Distribution of values of quadratic forms at integral points,, , (2010).

[23]

J. Griffin and J. Marklof, Limit theorems for skew translations,, J. Mod. Dyn. 8 (2014), 8 (2014), 177. doi: 10.3934/jmd.2014.8.177.

[24]

R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65. doi: 10.1090/S0273-0979-1982-15004-2.

[25]

G. H. Hardy and J. E. Littlewood, Some problems of diophantine approximation,, Acta Math., 37 (1914), 193. doi: 10.1007/BF02401834.

[26]

________, Some problems of diophantine approximation: An additional note on the trigonometrical series associated with the elliptic theta-functions,, Acta Math., 47 (1926), 189. doi: 10.1007/BF02544111.

[27]

A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in collaboration with E. A. Robinson, (1999), 107. doi: 10.1090/pspum/069/1858535.

[28]

________, Combinatorial Constructions in Ergodic Theory and Dynamics,, University Lecture Series, (2003). doi: 10.1090/ulect/030.

[29]

A. Katok and S. Katok, Higher cohomology for abelian groups of toral automorphisms,, Ergodic Theory Dynam. Systems, 15 (1995), 569. doi: 10.1017/S0143385700008531.

[30]

________, Higher cohomology for abelian groups of toral automorphisms. II. The partially hyperbolic case, and corrigendum,, Ergodic Theory Dynam. Systems, 25 (2005), 1909. doi: 10.1017/S0143385705000271.

[31]

H. Klingen, Introductory Lectures on Siegel Modular Forms,, Cambridge Studies in Advanced Mathematics, (1990). doi: 10.1017/CBO9780511619878.

[32]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces,, Invent. Math., 138 (1999), 451. doi: 10.1007/s002220050350.

[33]

A. Katok and V. Niţică, Rigidity in Higher Rank Abelian Group Actions. Volume I. Introduction and Cocycle Problem,, Cambridge Tracts in Mathematics, (2011). doi: 10.1017/CBO9780511803550.

[34]

A. Katok and F. Rodriguez Hertz, Rigidity of real-analytic actions of $SL(n,\mathbb Z)$ on $\mathbb T^n$: A case of realization of Zimmer program,, Discrete Contin. Dyn. Syst., 27 (2010), 609. doi: 10.3934/dcds.2010.27.609.

[35]

A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions,, Tr. Mat. Inst. Steklova, 216 (1997), 292.

[36]

J. C. Lagarias, Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators,, Trans. Amer. Math. Soc., 272 (1982), 545. doi: 10.2307/1998713.

[37]

G. W. Mackey, A theorem of Stone and von Neumann,, Duke Math. J., 16 (1949), 313. doi: 10.1215/S0012-7094-49-01631-2.

[38]

J. Marklof, Limit theorems for theta sums,, Duke Math. J., 97 (1999), 127. doi: 10.1215/S0012-7094-99-09706-5.

[39]

________, Theta sums, Eisenstein series, and the semiclassical dynamics of a precessing spin,, in Emerging Applications of Number Theory (Minneapolis, (1996), 405. doi: 10.1007/978-1-4612-1544-8_17.

[40]

J. Marklof, Pair correlation densities of inhomogeneous quadratic forms,, Ann. Math. (2), 158 (2003), 419. doi: 10.4007/annals.2003.158.419.

[41]

J. Moser, On commuting circle mappings and simultaneous Diophantine approximations,, Math. Z., 205 (1990), 105. doi: 10.1007/BF02571227.

[42]

D. Mumford, Tata Lectures on Theta. I,, With the collaboration of C. Musili, (1983). doi: 10.1007/978-1-4899-2843-6.

[43]

________, Tata Lectures on Theta. III,, With collaboration of M. Nori and P. Norman, (1991).

[44]

L. Schwartz, Théorie des Distributions,, Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1966).

[45]

N. A. Shah, Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds,, Duke Math. J., 148 (2009), 281. doi: 10.1215/00127094-2009-027.

[46]

C. L. Siegel, Symplectic Geometry,, Academic Press, (1964).

[47]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics,, Acta Math., 149 (1982), 215. doi: 10.1007/BF02392354.

[48]

R. Tolimieri, Heisenberg manifolds and theta functions,, Trans. Amer. Math. Soc., 239 (1978), 293. doi: 10.1090/S0002-9947-1978-0487050-7.

[49]

A. Weil, Sur certains groupes d'opérateurs unitaires,, Acta Math., 111 (1964), 143. doi: 10.1007/BF02391012.

[50]

T. D. Wooley, Perturbations of Weyl sums,, Int. Math. Res. Notices, (2015). doi: 10.1093/imrn/rnv225.

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