Equidistribution of expanding translates of curves and Dirichlet’s theorem on Diophantine approximation. (English) Zbl 1210.11083
Dirichlet found a simultaneous approximation result of the following shape. Given reals \(\alpha_1,\dots,\alpha_n\) and \(N\geq1\), there are integers \(x_1, \dots, x_n\), \(y\) (not all zero) with \(|\alpha_1x_1+\cdots+\alpha_nx_n+y|< N^{-n}\), and with \(\max(|x_1|,\dots,|x_n|)\leq N\). Dually, for each \(N\geq1\), there are integers \(x_1,\dots,x_n\), \(y\) (not all zero) with \({\max(|\alpha_1y-x_1|,\dots,|\alpha_ny-x_n|)}< N^{-1}\), and with \(|y|\leq N^n\). H. Davenport and W. M. Schmidt [Acta Arith. 16, 413–424 (1970; Zbl 0201.05501)] proved that each of these statements is false for almost every \((\alpha_1,\dots,\alpha_n)\) if the right-hand sides of the inequalities are multiplied by any number \(\mu < 1\). This remarkable sharpness – that Dirichlet’s theorems ‘cannot be \(\mu\)-improved’ for almost all points in \(\mathbb R^n\) – has several generalizations and extensions due to Baker, Dodson, Rynne, Vickers, Bugeaud, Kleinbock and Weiss. In particular, D. Kleinbock and B. Weiss [J. Mod. Dyn. 2, No. 1, 43–62 (2008; Zbl 1143.11022)] expressed the problem using the language of flows on homogeneous spaces and exploited results of S. G. Dani [J. Reine Angew. Math. 359, 55–89 (1985; Zbl 0578.22012); correction, J. Reine Angew. Math. 360, 214 (1985; Zbl 0578.22013)] and of D. Y. Kleinbock and G. A. Margulis [Ann. Math. (2) 148, No. 1, 339–360 (1998; Zbl 0922.11061)]. They proved that the Dirichlet theorems cannot be \(\mu\)-improved for almost all points lying on any nondegenerate curve in \(\mathbb R^n\) for some small \(\mu < 1\).
In this paper, Ratner’s theorem on ergodic properties of unipotent flows on homogeneous spaces is used to prove generalizations of the following form. Let \(\phi:[a,b] \rightarrow\mathbb R^n\) be an analytic curve whose image is not contained in any proper affine subspace. Then Dirichlet’s theorems cannot be improved (for any \(\mu\in(0,1)\)) for \(\phi(s)\) for almost all \(s\in [a,b]\). This is deduced from a more general result: Let \(\Lambda\) be a lattice in a Lie group \(L\), and let \(\rho:\text{SL}(n+1,\mathbb R) \rightarrow L\) be a continuous homomorphism; let \(x_0 \in L/\Lambda\) have the property that its orbit \(Hx_0\) under a minimal closed subgroup \(H\) of \(L\) containing the image of \(\rho\) is closed, and admits a finite \(H\)-invariant measure \(\mu_H\). Denote by \(a_t\) the diagonal matrix \(\text{diag}(e^{nt},e^{-t},e^{-t}, \dots, e^{-t})\) in \(\text{SL}(n+1,\mathbb R)\) for and by \(u(x_1, \dots,x_n)\) the unipotent matrix with first row \(1,x_1, \dots, x_n\) (and other rows as in the identity matrix). Then, for any bounded continuous \(f\) on \(L/\Lambda\), \[ \lim_{t \rightarrow \infty} \frac{1}{|b-a|} \int_a^b f(\rho(a_t u(\phi(s)))x_0)\,ds = \int_{Hx_0} f \,d \mu_H. \] An intermediate result (Proposition 4.2) describes how the linear actions of various copies of \(\text{SL}(2,\mathbb R)\) (whose action has been studied by the author [Duke Math. J. 148, No. 2, 281–304 (2009; Zbl 1171.37004)]) inside \({\text{SL}(n+1,\mathbb R)}\) interact.
In this paper, Ratner’s theorem on ergodic properties of unipotent flows on homogeneous spaces is used to prove generalizations of the following form. Let \(\phi:[a,b] \rightarrow\mathbb R^n\) be an analytic curve whose image is not contained in any proper affine subspace. Then Dirichlet’s theorems cannot be improved (for any \(\mu\in(0,1)\)) for \(\phi(s)\) for almost all \(s\in [a,b]\). This is deduced from a more general result: Let \(\Lambda\) be a lattice in a Lie group \(L\), and let \(\rho:\text{SL}(n+1,\mathbb R) \rightarrow L\) be a continuous homomorphism; let \(x_0 \in L/\Lambda\) have the property that its orbit \(Hx_0\) under a minimal closed subgroup \(H\) of \(L\) containing the image of \(\rho\) is closed, and admits a finite \(H\)-invariant measure \(\mu_H\). Denote by \(a_t\) the diagonal matrix \(\text{diag}(e^{nt},e^{-t},e^{-t}, \dots, e^{-t})\) in \(\text{SL}(n+1,\mathbb R)\) for and by \(u(x_1, \dots,x_n)\) the unipotent matrix with first row \(1,x_1, \dots, x_n\) (and other rows as in the identity matrix). Then, for any bounded continuous \(f\) on \(L/\Lambda\), \[ \lim_{t \rightarrow \infty} \frac{1}{|b-a|} \int_a^b f(\rho(a_t u(\phi(s)))x_0)\,ds = \int_{Hx_0} f \,d \mu_H. \] An intermediate result (Proposition 4.2) describes how the linear actions of various copies of \(\text{SL}(2,\mathbb R)\) (whose action has been studied by the author [Duke Math. J. 148, No. 2, 281–304 (2009; Zbl 1171.37004)]) inside \({\text{SL}(n+1,\mathbb R)}\) interact.
Reviewer: Thomas B. Ward (Norwich)
MSC:
| 11J83 | Metric theory |
| 22E40 | Discrete subgroups of Lie groups |
| 43A85 | Harmonic analysis on homogeneous spaces |
Citations:
Zbl 0201.05501; Zbl 1143.11022; Zbl 0578.22012; Zbl 0578.22013; Zbl 0922.11061; Zbl 1171.37004References:
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