Dynamics and Analytic Number Theory,
London Mathematical Society Lecture Note Series 437, eds. D. Badziahin, A. Gorodnik, N. Peyerimhoff, Cambridge University Press, 2016.
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Written by leading experts, this book explores several directions of current research at the interface between dynamics and analytic number theory. Topics include Diophantine approximation, exponential sums, Ramsey theory, ergodic theory and homogeneous dynamics. The origins of this material lie in the 'Dynamics and Analytic Number Theory' Easter School held at Durham University in 2014. Key concepts, cuttingedge results, and modern techniques that play an essential role in contemporary research are presented in a manner accessible to young researchers, including PhD students. This book will also be useful for established mathematicians. The areas discussed include ubiquitous systems and Cantortype sets in Diophantine approximation, flows on nilmanifolds and their connections with exponential sums, multiple recurrence and Ramsey theory, counting and equidistribution problems in homogeneous dynamics, and applications of thin groups in number theory. Both dynamical and 'classical' approaches towards number theoretical problems are also provided.
"Metric Diophantine approximation  aspects of recent work" by Victor Beresnevich, Felipe Ramírez and Sanju Velani
"Exponents of Diophantine approximation" by Yann Bugeaud
"Effective equidistribution of nilflows and bounds on Weyl sums" by Giovanni Forni
"Multiple recurrence and finding patterns in dense sets" by Tim Austin
"Diophantine problems and homogeneous dynamics" by Manfred Einsiedler and Tom Ward
"Applications of thin orbits" by Alex Kontorovich
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
nonamenable 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 M. Bjourklund)
Central limit theorems in the Geometry of Numbers
( PDF )
submitted.
 (with M. Bjourklund)
Central Limit Theorems for group actions which are exponentially mixing of all orders
( PDF )
submitted.
 (with M. Bjourklund and M. Einsiedler)
Quantitative multiple mixing
( PDF )
submitted.
 (with R. Spatzier)
Smooth factors of projective actions of higher rank lattices and rigidity
( PDF )
to appear in Geometry & Topology
 (with A. Nevo and G. Yehoshua)
Counting lattice points in norm balls on higher rank simple Lie groups
( PDF )
to appear in Math. Res. Lett.
 (with T. Browning)
Powerfree values of polynomials on symmetric varieties
( PDF )
to appear in Proc. London Math. Soc.
 (with P. Vishe)
Diophantine approxoimation for products of linear forms  logarithmic improvements
( PDF )
to appear in Trans. Amer. Math. Soc.
 (with S. Kadyrov)
Diophantine approximation with restricted numerrators and denominators on semisimple groups
( PDF )
J. Theor. Nombres Bordeaux 29 (2017), no. 1, 128.
 (with L. Luzzi and R. Vehkalahti)
Towards a complete DMT classification of division algebra codes
( PDF )
2016 IEEE International Symposium on Information Theory,
29932997.
 (with A. Ghosh and A. Nevo)
Diophantine approximation exponents on homogeneous varieties
( PDF )
Recent trends in ergodic theory and dynamical systems, 181200, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 2015.
 (with A. Ghosh and A. Nevo)
Best possible rates of distribution of dense lattice orbits in homogeneous spaces
( PDF )
J. Reine Angew. Math.
DOI 10.1515/crelle20160001.
 (with A. Nevo)
Quantitative ergodic theorems and their numbertheoretic applications
( PDF )
Bull. Amer. Math. Soc. 52 (2015), no. 1, 65113.
 (with R. Spatzier)
Mixing properties of Z^k actions on nilmanifolds
( PDF )
Acta Mathematica. 215:1 (2015).

Lie groups, algebraic groups, and lattices
( PDF )
to appear in Proceedings of Bedlewo School.
 (with R. Spatzier)
Exponential mixing of nilmanifold automorphisms
( PDF )
Journal d'Analyse Mathematiques 123 (2014), 355–396.
 (with F. Paulin) Counting orbits of integral points in families of affine
homogeneous varieties and diagonal flows
( PDF )
Journal of Modern Dynamics 8 (2014), no. 1, 2559.
 (with A. Nevo)
Ergodic theory and the duality principle on homogeneous spaces
( PDF )
Geom. Funct. Anal. 24 (2014), no. 1, 159–244.
 (with A. Ghosh and A. Nevo)
Metric Diophantine approximation on homogeneous varieties
( PDF )
Compos. Math. 150 (2014), no. 8, 14351456.
 (with F. Ramirez)
Limit theorems for rankone groups
( PDF )
Proc. Amer. Math. Soc. 142 (2014), no. 4, 13591369.
 (with U. Bader, A. Furman, and B. Weiss)
Rigidity of group actions on homogeneous spaces III
( PDF )
Duke Math. J. 164 (2015), no. 1, 115155.
 (with R. TaklooBighash and Y. Tschinkel)
Multiple Mixing for adele groups and rational points
( PDF )
Eur. J. Math. 1 (2015), no. 3, 441461.
 (with S. Kadyrov) Algebraic numbers, hyperbolicity, and density modulo one
( PDF )
J. Number Theory 132 (2012), 24992509.
 (with A. Nevo) Splitting fields of elements in arithmetic groups
( PDF )
Math. Res. Lett. 18 (2011), no. 06, 12811288.
 (with A. Nevo) Lifting, restricting and sifting integral points on
affine homogeneous varieties
( PDF )
Compositio Math. 148 (2012), no. 6, 1695–1716.
 (with A. Nevo) On Arnold's and Kazhdan's equidistribution problems
( PDF )
Ergodic Theory Dynam. Systems 32 (2012), no. 6, 19721990.
 (with A. Ghosh and A. Nevo) Diophantine approximation and automorphic spectrum
( PDF )
Int. Math. Research Notices 2013, no. 21, 5002–5058.
 (with A. Nevo) Counting lattice points
( PDF )
J. Reine Angew. Math. 663 (2012), 127176.
 (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, 357380.
 (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, 509540.
 (with M. Borovoi and H. Oh) Rational points on homogeneous varieties and equidistribution of Adelic periods
( PDF )
Geom. Funct. Anal. 21 (2011), no. 2, 319392.
 (with H. Oh and N. Shah) Strong wavefront lemma and counting lattice points in sectors
( PDF )
Israel J. Math. 176 (2010), 419444.
 (with H. Oh and N. Shah) Integral points on symmetric varieties and Satake compactifications
( PDF )
Amer. J. Math.
131 (2009), 157.
 (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), 4797.
 (with V. Bergelson) Ergodicity and mixing of noncommuting epimorphisms
( PDF )
Proc. London Math. Soc. 95 (2007), 329359.
 Open problems in dynamics and related fields.
( PDF )
Journal of Modern Dynamics 1 (2007), no. 1, 135.
 (with B. Weiss) Distribution of lattice orbits on homogeneous varieties.
( DVI PDF )
Geom. Funct. Anal. 17 (2007), no. 1, 58115.
 (with F. Maucourant) Proximality and equidistribution on the Furstenberg boundary.
( DVI PDF )
Geom. Dedicata 113 (2005), 197213.
 (with H. Oh) Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary.
( DVI PDF )
Duke Math. J. 139 (2007), no. 3, 483525.
 (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, 549589.
 Oppenheim conjecture for pairs consisting of a linear form and a quadratic form.
( DVI PDF )
Trans. Amer. Math. Soc. 356 (2004), no. 11, 44474463.
 On Oppenheimtype conjecture for systems of quadratic forms.
( DVI PDF )
Israel J. Math. 140 (2004), 125144.
 Lattice action on the bounday of SL(n,R).
( DVI PDF )
Ergodic Theory Dynam. Systems 23 (2003), no. 6, 18171837.
 Local nearrings with commutative groups of units.
( abstract DVI )
Houston J. Math. 25 (1999), no. 2, 223234.
 The product of an abelian group and a nilpotent group.
Ukrainian Math. J. 51 (1999), no. 9, 13141320.
 On the product of groups with finite commutants.
in Contemporary problems of mechanics and mathematics, Lviv, 1998.
Talks
 Lectures on Diophantine approximation and Dynamics
(Summer School "Analytic Questions in Arithmetic", TIFR, Mumbai, India)
 GreenTao proof on arithmetic progressions in primes I. ( PDF )
 GreenTao 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 )
Notes
 GreenTao proof on arithmetic progressions in primes.
( PDF )