A generalization of Dirichlet approximation theorem for the affine actions on real line |
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Authors: | Mohammad Javaheri |
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Institution: | Department of Mathematics, University of Oregon, Eugene, OR 97403, USA |
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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 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|<C(t1/|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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Keywords: | Dirichlet approximation Affine actions Prime approximation |
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