An inhomogeneous Jarník theorem
A classical result of \textit{A. Khintchine} [M. Z. 24, 706--714 (1926; JFM 52.0183.02)] asserts that given a non-increasing function \(\psi: \mathbb Z_{>0}\to\mathbb R_{>0}\), the set \(K(\psi)= \{\xi\in\mathbb R:|q\xi- p|< \psi(q)\), for infinitely many integers \(p\) and \(q\) with \(q\geq 1\}\) has full Lebesgue measure if the sum \(\sum_{q\geq 1} \psi(q)\) diverges, while it has Lebesgue measure zero otherwise. However, one may search for a more precise statement in the convergence case because Hausdorff measure allows to discriminate between null sets. For instance, \textit{V. Jarník} [Rec. Math. Moscou 36, 371--382 (1929; JFM 55.0719.01)] and Besicovitch (1934) proved independently, that for any \(\lambda\geq 1\) the Hausdorff dimension of \(K(x\mapsto x^{-\lambda})\) is equal to \(2/(\lambda+ 1)\). Jarník obtained a more general and precise result in [Math. Z. 33, 05--543 (1931; Zbl 0001.32403)] which involves generalized Hausdorff measures. It also deals with simultaneous Diophantine approximation. In this paper, the generalized Hausdorff measure of sets of points in \(\mathbb R^s\) which satisfy an inhomogeneous system of Diophantine inequalities infinitely often has been computed. This proves an inhomogeneous analogue of the classical result of Jarník on simultaneous Diophantine approximation.
scientific article; zbMATH DE number 2112870