Generalizations of a result of Jarnik on simultaneous approximation

Abstract: Consider a non-increasing function $\Psi$ from the positive reals to the positive reals with decay $o(1/x)$ as $x$ tends to infinity. Jarnik proved in 1930 that there exist real numbers $\zeta_{1},...,\zeta_{k}$ together with $1$ linearly independent over $\mathbb{Q}$ with the property that all $q\zeta_{j}$ have distance to the nearest integer smaller than $\Psi(q)$ for infinitely many positive integers $q$, but not much smaller in a very strict sense. We give an effective generalization of this result to the case of successive powers of real $\zeta$. The method also allows to generalize corresponding results for $\zeta$ contained in special fractal sets such as the Cantor set.