Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions
We consider the distribution of the orbits of the number 1 under the -transformations as varies. Mainly, the size of the set of for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set is determined, where is a given point in and is a sequence of integers tending to infinity as . For the proof of this result, the
