Simultaneous Diophantine approximation on the circle and Hausdorff dimension (Q2725003)

Let \(M\) be an \(m\)-dimensional manifold in \(\mathbb R^n\) and let \(\mathcal S_v(M)\) denote the set of \(\mathbf x=(x_1,\dots,x_n)\in M\) such that \(\max|q x_i|<q^{-v}\) is true for infinitely many positive integers \(q\). A basic result of \textit{D. Y. Kleinbock} and \textit{G. A. Margulis} [Ann. Math. (2) 148, 339-360 (1998; Zbl 0922.11061)] states that the Lebesgue measure of \(\mathcal S_v(M)\) induced on \(M\) is zero for \(v>1/n\) if \(M\) is sufficiently smooth and satisfies a non-degeneracy condition. Take \(M\) to be the unit circle \(\mathbb S^1\) in \(\mathbb R^2\). \textit{Yu. V. Mel'nichuk} [Dopov. Akad. Nauk. Ukr. RSR, Ser. A 1978, 792-796 (1978; Zbl 0387.10021); Russian translation in Dokl. Akad. Nauk Ukr. SSR, Ser. A 1978, No. 9, 793-796 (1978)] has proved that for the Hausdorff dimension we have NEWLINE\[NEWLINE\tfrac 12(1+v)^{-1}\leq\dim\mathcal S_v(\mathbb S^1)\leq(1+v)^{-1} \quad \text{if }v>1.NEWLINE\]NEWLINE The present authors close the gap by showing that NEWLINE\[NEWLINE\dim\mathcal S_v(\mathbb S^1)=(1+v)^{-1}\quad\text{for }v>1.NEWLINE\]NEWLINE A proof of Mel'nichuk's upper bound is also included for the sake of completeness.