Dynamics and Analytic Number Theory, London Mathematical Society Lecture Note Series 437, eds. D. Badziahin, A. Gorodnik, N. Peyerimhoff, Cambridge University Press, 2016.

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, cutting-edge 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 Cantor-type 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 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 M. Fraczyk and A. Nevo) Automorphic density estimates and optimal Diophantine exponents ( PDF )

(with M. Björklund and R. Fregoli) Decorrelation estimates for translated measures under diagonal flows ( PDF )

(with J. Li and C. Sert) Stationary measures for SL2(R)-actions on homogeneous bundles over flag varieties ( PDF )

(with M. Björklund and R. Fregoli) A uniform metrical theorem in multiplicative Diophantine approximation ( PDF )

(with A. Nevo) On discrepancy, intrinsic Diophantine approximation, and spectral gaps ( PDF )

(with M. Björklund) Poisson approximation and Weibull asymptotics in the geometry of numbers ( PDF )

(with M. Björklund) Effective multiple equidistribution of translated measures ( PDF )

(with A. Nevo) Discrepancy of rational points in simple algebraic groups ( PDF )

(with M. Björklund) Counting in generic lattices and higher rank actions ( PDF )

(with A. Ghosh and A. Nevo) Counting intrinsic Diophantine approximations in simple algebraic groups ( PDF )

(with C. Daw, E. Ullmo, and J. Li) The space of homogeneous probability measures on the Satake compactification is compact ( PDF )

(with C. Daw and E. Ullmo) Convergence of measures on compactifications of locally symmetric spaces ( PDF )

Higher-order correlations for group actions ( PDF )
to appear in Tata Institute Conference Proceedings .

(with M. Bjourklund) Central Limit Theorems for Diophantine approximants ( PDF )

(with A. Ghosh and A. Nevo) Optimal density for values of generic polynomial maps ( PDF )

(with M. Bjourklund) Central Limit Theorems for group actions which are exponentially mixing of all orders ( PDF )
accepted to Journal d'Analyse Mathematiques.

(with R. Spatzier) Smooth factors of projective actions of higher rank lattices and rigidity ( PDF )
Geom. Topol. 22 (2018), no. 2, 1227-1266.

(with P. Vishe) Diophantine approxoimation for products of linear forms - logarithmic improvements ( PDF )
Trans. Amer. Math. Soc. 370 (2018), no. 1, 487-507.

(with M. Bjourklund and M. Einsiedler) Quantitative multiple mixing ( PDF )
to appear in J. Eur. Math. Soc. (JEMS)

(with M. Bjourklund) Central limit theorems in the Geometry of Numbers ( PDF )
Electron. Res. Announc. Math. Sci. 24 (2017), 110-122.

(with A. Nevo and G. Yehoshua) Counting lattice points in norm balls on higher rank simple Lie groups ( PDF )
Math. Res. Lett. 24 (2017), no. 5, 1285-1306.

(with T. Browning) Power-free values of polynomials on symmetric varieties ( PDF )
Proc. Lond. Math. Soc. 114 (2017), no. 6, 1044-1080.

(with S. Kadyrov) Diophantine approximation with restricted numerrators and denominators on semisimple groups ( PDF )
J. Theor. Nombres Bordeaux 29 (2017), no. 1, 1-28.

(with L. Luzzi and R. Vehkalahti) Towards a complete DMT classification of division algebra codes ( PDF )
2016 IEEE International Symposium on Information Theory, 2993-2997.

(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/crelle-2016-0001.

(with A. Ghosh and A. Nevo) Diophantine approximation exponents on homogeneous varieties ( PDF )
Recent trends in ergodic theory and dynamical systems, 181-200, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 2015.

(with A. Nevo) Quantitative ergodic theorems and their number-theoretic applications ( PDF )
Bull. Amer. Math. Soc. 52 (2015), no. 1, 65-113.

(with R. Spatzier) Mixing properties of Z^k actions on nilmanifolds ( PDF )
Acta Mathematica. 215:1 (2015).

(with U. Bader, A. Furman, and B. Weiss) Rigidity of group actions on homogeneous spaces III ( PDF )
Duke Math. J. 164 (2015), no. 1, 115-155.

(with R. Takloo-Bighash and Y. Tschinkel) Multiple Mixing for adele groups and rational points ( PDF )
Eur. J. Math. 1 (2015), no. 3, 441-461.

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, 25-59.

(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, 1435-1456.

(with F. Ramirez) Limit theorems for rank-one groups ( PDF )
Proc. Amer. Math. Soc. 142 (2014), no. 4, 1359-1369.

(with A. Ghosh and A. Nevo) Diophantine approximation and automorphic spectrum ( PDF )
Int. Math. Research Notices 2013, no. 21, 5002-5058.

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

(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, 1972-1990.

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

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

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

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

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

(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.

Talks

Lectures on Diophantine approximation and Dynamics (Summer School "Analytic Questions in Arithmetic", TIFR, Mumbai, India)

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 )

Notes

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