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Diophantine problems and homogeneous dynamics. (English) Zbl 1412.37001

Badziahin, Dzmitry (ed.) et al., Dynamics and analytic number theory. Proceedings of the Durham Easter School, Durham, UK, March 31 – April 4, 2014. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 437, 258-288 (2016).
This is a survey paper, which gives an introduction to homogeneous dynamics, problems of lattice-point counting, and Diophantine approximations. The paper starts with the application of an equidistribution result to the Gauss circle problem. The authors continue by introducing the upper half-plane model of the hyperbolic plane \(\mathbb{H}\). They show how \(\mathrm{SL}_2(\mathbb{R})\) and \(\mathrm{PSL}_2(\mathbb{R})\) act on \(\mathbb{H}\), and establish some key properties of this action. Next they prove an asymptotic result on lattice-point counting in \(\mathrm{PSL}_2(\mathbb{Z})\). The exposition in the present paper follows the dynamical approach in the work of A. Eskin and C. McMullen [Duke Math. J. 71, No. 1, 181–209 (1993; Zbl 0798.11025)] and W. Duke et al. [Duke Math. J. 71, No. 1, 143–179 (1993; Zbl 0798.11024)]. Finally, they show how Dirichlet’s classical theorem on Diophantine approximation can be proved by homogeneous dynamics via the Dani correspondence, and how this approach generalizes to Diophantine approximation problems on curved smooth manifolds.
For the entire collection see [Zbl 1362.11002].

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37A17 Homogeneous flows
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11P21 Lattice points in specified regions
11J99 Diophantine approximation, transcendental number theory
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