Some applications of homogeneous dynamics to number theory. (Q2778790)

The aim of this paper is to describe several principles responsible for a particular class of applications of flows in the space of lattices to number theory. Consider the following two Diophantine problems:NEWLINENEWLINE (i) Given a non degenerate indefinite quadratic form \(Q\) of signature \((m,n)\), study the set of its values at integer points.NEWLINENEWLINE (ii) Given \(m\) vectors \(y_1, \dots,y_m\) in \(\mathbb{R}^n\), how small (simultaneously) can be the values of \(|y_iq+p_i|\), \(p_i\) in \(\mathbb{Z}\) when \(q=(q_1,\dots,q_n)\) in \(\mathbb{Z}^n\) is far from \(0\)?NEWLINENEWLINE One possible approach to problem (i) is to write \(Q(X)=\lambda S_{m,n}(gx)\) where \(\lambda\in\mathbb{R}\), \(g\in\text{SL}_k(\mathbb{R})\) and \(S_{m,n}(x_1,\dots,x_k) =x_1^2+ \cdots+x_m^2-x^2_{m+1}-\cdots-x^2_k\). Then the problem reduces to studying values of the standard form \(S_{m,n}\) applied to the collection of the vectors of the form \((gx\mid x\in Z^k)\). And the dynamical approach consists in studying the action of the stabilizer of the form \(S_{m,n}\) on such collections.NEWLINENEWLINE To approach problem (ii), one can put together \(y_1q+ p_1,\dots,y_mq+p_m\) and \(q_1, \dots,q_n\) and consider the lattice \(L_Y\mathbb{Z}^{m+n}\) where \(L_Y=\left( \begin{smallmatrix} I_m & Y\\ 0 & I_n \end{smallmatrix}\right)\) and \(Y\) is the matrix with rows \(y_1^t,\dots, y_m^t\). In this case the orbit of the lattice under a certain group action provides a way to study Diophantine properties of \(y_1,\dots,y_m\). In both cases the initial data of a number theoretic problem is used to construct a lattice in a Euclidean space and work with the collection of all such lattices. Several subproblems of problem (ii) and recent results obtained by means of homogeneous dynamics have been considered. Multiplicative Diophantine approximation has also been discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].