The very active domain of Diophantine approximation on manifolds is mainly devoted to the study of the approximation of points in a manifold \(X\) by rational points in the ambient space. Here, the authors consider intrinsic Diophantine approximation, where the approximating points lie in the manifold \(X\) itself. The example which is considered in the paper under review is where \(X\) is the unit sphere \(S^n\) in \({\mathbb{R}}^{n+1}\).
The first result is an analog of Dirichlet’s theorem. For any \(n\geq 1\) there exists a constant \(C_n>0\) such that for any \(\alpha\in S^n\) there exist infinitely many \({\mathbf {p}}/q\) in \(S^n\) (\(\mathbf {p} \in \mathbb Z^{n+1}\) primitive, \(q \in \mathbb N\)) with \[ \left| \alpha-\frac{{\mathbf {p}}}{q}\right|<\frac{C_n}{q}\cdotp \] The authors show that \(C_n\) cannot be replaced by an arbitrary small constant : they define the set of badly approximable points on \(S^n\) as the set \({\mathrm{BA}}(S^n)\) of \(\alpha\in S^n\) such that there exists \(c=c(\alpha)>0\) such that, for all \({\mathbf {p}}/q\in S^n\cap {\mathbb {Q}}^{n+1}\), \[ \left| \alpha-\frac{{\mathbf {p}}}{q}\right|>\frac{c}{q}\cdotp \] They show that this set \({\mathrm{BA}}(S^n)\) is thick, meaning that its intersection with any non empty open subset of \(S^n\) has full Hausdorff dimension.
They also prove an analog of Khinchine theorem for intrinsic approximation; in the special case that they consider, they obtain stronger results than the previous ones in [A. Ghosh et al., Compos. Math. 150, No. 8, 1435–1456 (2014;Zbl 1309.37005)]. Also, using the notion of mass transference developed in [V. Beresnevich andS. Velani, Ann. Math. (2) 164, No. 3, 971–992 (2006;Zbl 1148.11033)], the authors obtain refined results involving the Hausdorff measure.
The main tool is a correspondence between approximation and dynamics which is described explicitly here and which is also related with the works ofD. Y. Kleinbock andG. A. Margulis [Invent. Math. 138, No. 3, 451–494 (1999;Zbl 0934.22016)] andC. Druţu [Math. Ann. 333, No. 2, 405–470 (2005;Zbl 1082.11047)].

Reviewer: Michel Waldschmidt (Paris)

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