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

Abstract

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 T 0 ( x ) = x / a and T 1 ( x ) = b x + 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 T 0 and T 1 such that | γ ( x ) − u | < C ( t 1 / | γ | − 1 ) , where C and t are constants independent of γ, and | γ | is the length of γ as a word in the semigroup. Finally, we discuss the problem of approximating an arbitrary real number by the ratios of prime numbers and the ratios of logarithms of prime numbers.