A. Gorodnik and A. Nevo, The ergodic theory of lattice subgroups, Annals of Mathematics Studies 172, Princeton University Press, 2010.  

The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. We develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases.

Research articles

• (with R. Spatzier) Mixing properties of Z^k actions on nilmanifolds ( PDF )
  submitted, 2012.


• Lie groups, algebraic groups, and lattices ( PDF )
  submitted, 2012.


• (with R. Spatzier) Exponential mixing of nilmanifold automorphisms ( PDF )
  submitted, 2012.


• (with A. Nevo) Ergodic theory and the duality principle on homogeneous spaces ( PDF )
  submitted, 2012.


• (with A. Ghosh and A. Nevo) Metric Diophantine approximation on homogeneous varieties ( PDF )
  submitted, 2012.


• (with F. Ramirez) Limit theorems for rank-one groups ( PDF )
  to appear in Proc. Amer. Math. Soc.


• (with S. Kadyrov) Diophantine approximation with restricted numerrators and denominators on semisimple groups ( PDF )
  submitted, 2012.


• (with U. Bader, A. Furman, and B. Weiss) Rigidity of group actions on homogeneous spaces III ( PDF )
  submitted, 2012.


• (with R. Takloo-Bighash and Y. Tschinkel) Multiple Mixing for adele groups and rational points ( PDF )
  submitted, 2011.


• (with F. Paulin) Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows ( PDF )
  submitted, 2011.


• (with S. Kadyrov) Algebraic numbers, hyperbolicity, and density modulo one ( PDF )
  J. Number Theory 132 (2012), 2499-2509.


• (with A. Nevo) Splitting fields of elements in arithmetic groups ( PDF )
  Math. Res. Lett. 18 (2011), no. 06, 1281-1288.


• (with A. Nevo) Lifting, restricting and sifting integral points on affine homogeneous varieties ( PDF )
  Compositio Math. 2012, doi:10.1112/S0010437X12000516.


• (with A. Nevo) On Arnold's and Kazhdan's equidistribution problems ( PDF )
  Ergodic Theory Dynam. Systems 32 (2012), no. 6, 1972-1990.


• (with A. Ghosh and A. Nevo) Diophantine approximation and automorphic spectrum ( PDF )
  Int. Math. Research Notices 2012, doi:10.1093/imrn/rns198.


• (with A. Nevo) Counting lattice points ( PDF )
  J. Reine Angew. Math. 663 (2012), 127-176.


• (with V. Bergelson) Trieste lectures on ergodic theorems and applications ( PDF )
  preprint, 2008.


• (with N. Shah) Khinchin theorem for integral points on quadratic varieties ( PDF )
  Math. Ann. 350 (2011), no. 2, 357-380.


• (with T. Hitchman and R. Spatzier) Regularity of conjugacies of algebraic actions of Zariski dense groups ( PDF )
  Journal of Modern Dynamics 2 (2008), no. 3, 509-540.


• (with M. Borovoi and H. Oh) Rational points on homogeneous varieties and equidistribution of Adelic periods ( PDF )
  Geom. Funct. Anal. 21 (2011), no. 2, 319-392.


• (with H. Oh and N. Shah) Strong wavefront lemma and counting lattice points in sectors ( PDF )
  Israel J. Math. 176 (2010), 419-444.


• (with H. Oh and N. Shah) Integral points on symmetric varieties and Satake compactifications ( PDF )
  Amer. J. Math. 131 (2009), 1-57.


• (with F. Maucourant and H. Oh) Manin's conjecture on rational points of bounded height and adelic mixing ( PDF )
  Annales Scientifiques de l'Ecole Normale Superieure 41 (2008), 47-97.


• (with V. Bergelson) Ergodicity and mixing of noncommuting epimorphisms ( PDF )
  Proc. London Math. Soc. 95 (2007), 329-359.


• Open problems in dynamics and related fields. ( PDF )
  Journal of Modern Dynamics 1 (2007), no. 1, 1-35.


• (with B. Weiss) Distribution of lattice orbits on homogeneous varieties. ( DVI     PDF )
  Geom. Funct. Anal. 17 (2007), no. 1, 58-115.


• (with F. Maucourant) Proximality and equidistribution on the Furstenberg boundary. ( DVI     PDF )
  Geom. Dedicata 113 (2005), 197-213.


• (with H. Oh) Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary. ( DVI     PDF )
  Duke Math. J. 139 (2007), no. 3, 483-525.


• (with V. Bergelson) Weakly mixing group actions: a brief survey and an example. ( DVI     PDF )
  in Modern Dynamical Systems and Applications, B. Hasselblatt, M. Brin, Y. Pesin, eds., Cambridge University Press, New York, 2004.


• Uniform distribution of orbits of lattices on spaces of frames. ( DVI     PDF )
  Duke Math. J. 122 (2004), no. 3, 549-589.


• Oppenheim conjecture for pairs consisting of a linear form and a quadratic form. ( DVI     PDF )
  Trans. Amer. Math. Soc. 356 (2004), no. 11, 4447-4463.


• On Oppenheim-type conjecture for systems of quadratic forms. ( DVI     PDF )
  Israel J. Math. 140 (2004), 125-144.


• Lattice action on the bounday of SL(n,R). ( DVI     PDF )
  Ergodic Theory Dynam. Systems 23 (2003), no. 6, 1817-1837.


• Local near-rings with commutative groups of units. ( abstract     DVI )
  Houston J. Math. 25 (1999), no. 2, 223-234.


• The product of an abelian group and a nilpotent group.
  Ukrainian Math. J. 51 (1999), no. 9, 1314-1320.


• On the product of groups with finite commutants.
  in Contemporary problems of mechanics and mathematics, Lviv, 1998.

• Lectures on Diophantine approximation and Dynamics (Summer School "Analytic Questions in Arithmetic", TIFR, Mumbai, India)
  • Lecture I: Dani Correspondence: Diophantine Approximation and Flows
  • Lecture II: Khinchin Theorem and Exponential Mixing
  • Lecture III: Schmidt Games, Badly Approximable Numbers, and Bounded Geodesics
  • Lecture IV: Diophantine Approximation on Quadratic Surfaces and Ramanujan Conjecture


• Green-Tao proof on arithmetic progressions in primes I. ( PDF )


• Green-Tao proof on arithmetic progressions in primes II. ( PDF )


• (with Barak Weiss) Uniform distribution of orbits of lattices. ( PDF )


• (with Barak Weiss) Equidistribution on homogeneous spaces. ( PDF )


• (with Amos Nevo) Ergodic Theory of semisimple lattices. ( PDF )


• (with Francois Maucourant and Hee Oh) Rational points on algebraic varieties and homogeneous flows. ( PDF )

• Green-Tao proof on arithmetic progressions in primes. ( PDF )