On simultaneous rational approximation to a real number and its integral powers, II

2019/06/13 by Dzmitry Badziahin, Yann Bugeaud, Badziahin, Dmitry +1
Mathematics · #11J13 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1906.05508

openalex publication_date 2019/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a positive integer n and a real number ξ, let λ_n (ξ) denote the supremum of the real numbers λ for which there are arbitrarily large positive integers q such that || q ξ||, || q ξ² ||, … , ||q ξⁿ|| are all less than q^-λ. Here, || ⋅ || denotes the distance to the nearest integer. We establish new results on the Hausdorff dimension of the set of real numbers ξ such that λ_n (ξ) is equal (or greater than or equal) to a given value.

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