Simultaneous Diophantine approximation on manifolds and Hausdorff dimension
Let \(M\) be a \(C^k\) manifold in \(\mathbb R^n\) for any \(k,n\in\mathbb N\), and let \(\mathcal S_\tau(M)\) denote the set of \(\mathbf x=(x_1,\dots,x_n)\in M\) such that \(\max\|qx_i\|<q^{-\tau}\) is true for infinitely many integers \(q\). Take \(n=2\) and \(\Gamma_l=\{(x_1,x_2)\in \mathbb R^2 : x_1^l+x_2^l=1\}\) where \(l\in\mathbb Z\) and \(l\geq 2\). \textit{H. Dickinson} and \textit{M. M. Dodson} [Math. Proc. Camb. Philos. Soc. 130, 515--522 (2001; Zbl 0992.11046)] proved that the Hausdorff dimension \(\dim\mathcal S_\tau(\Gamma_2) = (1+\tau)^{-1}\) for \(\tau>1\). On the other hand, it is proved in the book of \textit{V. I. Bernik} and \textit{M. M. Dodson} [Metric Diophantine approximation on manifolds. Cambridge Tracts in Mathematics 137 (1999; Zbl 0933.11040)] that \(\dim\mathcal S_\tau(\Gamma_l)=0\) for \(l\geq 3\) and \(\tau>l-1\). Therefore, the condition that the curvature be nonzero everywhere does not distinguish between curves for which the dimension is positive and those for which it is zero, for large \(\tau\). Here the author shows that something similar is true in general, i.e., for any \(n\) and any \(C^k\) manifold \(M\), there exist \(C^k\) manifolds \(M_{\text{z}}\), \(M_{\text{p}}\) arbitrarily ''\(C^k\)-close'' to \(M\) such that for every sufficiently large \(\tau\), \(\dim\mathcal S_\tau(M_{\text{z}})=0\), \(\dim\mathcal S_\tau(M_{\text{p}})>0.\)
scientific article; zbMATH DE number 1902863
scientific article