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Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms. (English) Zbl 1200.11055

H. Davenport and W. M. Schmidt [Acta Arith. 16, 413–424 (1970; Zbl 0201.05501)] said that Dirichlet’s theorem – for any \(\xi\in\mathbb R^k\) and \(N\in\mathbb N^k\) the inequalities \(|\sum_{i=1}^{k}q_i\xi_i-p|\leq (N_1\cdots N_k)^{-1}\), \(| q_i|<N_i\) for all \(i\), have non-trivial integral solutions – cannot be improved along an infinite set \(\mathcal{N}\subset\mathbb Z^k\) if for every \(\mu\in(0,1)\) there are infinitely many \(N\in\mathcal N\) for which the inequalities \(|\sum_{i=1}^{k}q_i\xi_i-p|\leq \mu(N_1\cdots N_k)^{-1}\), \(| q_i|<\mu N_i\) for all \(i\) have no integral solutions, and showed that Dirichlet’s theorem cannot be improved along the diagonal set \(\{(N,N,\dots,N)\}\).
D. Kleinbock and B. Weiss [J. Mod. Dyn. 2, No. 1, 43–62 (2008; Zbl 1143.11022)] related these questions to the dynamics of flows on homogeneous spaces, and extended the results to sets \(\mathcal{N}\) with infinite projection onto each coordinate. For almost every vector \(\xi\) constrained to lie on specific curves or submanifolds, Kleinbock and Weiss found similar results for small values of \(\mu\).
For \(\mathcal{N}\) the diagonal and \(\xi\) lying on an analytic curve that is not contained in a proper affine subspace, the author [Invent. Math. 177, No. 3, 509–532 (2009; Zbl 1210.11083)] showed that Dirichlet’s theorem cannot be improved for almost every \(\xi\) for \(\mu\in(0,1)\). Here this result is extended to allow any infinite set \(\mathcal{N}\). The approach is to formulate the problem in terms of the asymptotic distribution of translates of curves on the homogeneous space \(\text{SL}_n(\mathbb R)/\text{SL}_n(\mathbb Z)\) under certain flows, and to use ergodic methods to establish equidistribution. There are many technical difficulties to be overcome in adapting the strategy adopted in the diagonal case to the more general setting.

MSC:

11J83 Metric theory
22E40 Discrete subgroups of Lie groups
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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