A generalization of Dirichlet approximation theorem for the affine actions on real line

In this paper, we obtain strong density results for the orbits of real numbers under the action of the semigroup generated by the affine transformations T0(x)=x/a and T1(x)=bx+1, where a,b>1. These density results are formulated as generalizations of the Dirichlet approximation theorem and improve the results of Bergelson, Misiurewicz, and Senti. We show that for any x,u>0 there are infinitely many elements γ in the semigroup generated by T0 and T1 such that |γ(x)−u|