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, 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