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). Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[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. Google Scholar

[19]

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

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[31]

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

[32]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

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

[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. Google Scholar

[40]

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

[41]

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

[42]

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

[43]

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

[44]

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

[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. Google Scholar

[46]

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

[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. Google Scholar

[48]

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

[49]

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

[50]

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

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). Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[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. Google Scholar

[19]

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

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[31]

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

[32]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

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

[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. Google Scholar

[40]

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

[41]

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

[42]

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

[43]

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

[44]

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

[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. Google Scholar

[46]

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

[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. Google Scholar

[48]

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

[49]

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

[50]

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

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