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.

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