Hausdorff dimension for the set of points connected with the generalized\n Jarn 'ik-Besicovitch set

2019/11/28 by Ayreena Bakhtawar, Bakhtawar, Ayreena
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1911.12550

openalex publication_date 2019/11/28 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

In this article we aim to investigate the Hausdorff dimension of the set of\npoints x \∈ [0,1) such that for any r\∈\ℕ, beginalign*\na_n+1(x)a_n+2(x)\⋯ a_n+r(x)\≥≠^\τ(x)(h(x)+\⋯+h(T^n-1(x))) align* holds for infinitely many\nn\∈\ℕ, where h and \τ are positive continuous functions, T\nis the Gauss map and a_n(x) denote the nth partial quotient of x in its\ncontinued fraction expansion. By appropriate choices of r, \τ(x) snd\nh(x) we obtain the classical Jarn 'ik-Besicovitch Theorem as well as more\nrecent results by Wang-Wu-Xu, Wang-Wu, Huang-Wu-Xu and\nHussain-Kleinbock-Wadleigh-Wang.\n

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