2023/06/21 by Yan Huang, Huang, Yan, Charlene Kalle +1
Mathematics · #Approximation Theory and Sequence Spaces #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2306.12114
openalex publication_date 2023/06/21 · openalex created_date 2023/06/24 · openalex updated_date 2026/07/28
For a fixed α, each real number x ∈ (0,1) can be represented by many different generalised α-Lüroth expansions. Each such expansion produces for the number x a sequence of rational approximations ((pn)/(qn))n ≥ 1. In this paper we study the corresponding approximation coefficients (θn(x))n ≥ 1, which are given by θn (x): = qn |x-(pn)/(qn)|. We give the cumulative distribution function and the expected average value of the θn and we identify which generalised α-Lüroth expansion gives the best approximation properties. We also analyse the structure of the set \mathcal Mα of possible values that the expected average value of θn can take, thus answering a question from \citeBarrionuevo-Burton-Dajani-Kraaikamp-1994.