The Asymptotic Behaviour Of A Class Of Linear Homogeneous Cantor-integers

This thesis mainly studies the asymptotic behavior of a class of linear homogeneous Cantor integers {an}n≥0.The Cantor integers {an}n≥0 are characterized by the relationship between the digits of the terms an and the indices n in the corresponding expansions in different integer bases.In precise,for given integers p>s≥2,if there is a strictly increasing function h:{0,1,…,s-1}→ {0,1,…,p-1} such that for any nonnegative integer n,the s-ary expansion of n is n=[εk,…,ε0]s implies that the p-ary expansion of an is an=[h(εk),…,h(ε0)]p,then an is called a Cantor integer.The linearly homogeneous means that h is a linearly homogeneous function.This thesis considers exactly the case s=[p/2],h=2x.Including the first chapter of introduction and the second chapter of preliminary,there are five chapters in the thesis.This thesis starts from the p-uniform morphism σ:σ(1)=(10)p/2,σ(0)=(00)p/2,and then illustrates that the characteristic sequence of 1 in the fixed point c={cn}n≥0 ofσ is exactly the Cantor integer sequence in the case of s=[p/2],h=2x.In the third chapter,we analyze the growth rate of an,and get the fact that the growth order of an is logsp.Furthermore we analyze the distribution of {an/nlogsp}n≥1,and get the conclusion that{an/nlogsp}n≥1 is dense in[mp,2],where mp=(p-2)/(p-1)if p is even otherwise 1.In view of the inseparable relationship between Cantor integers,Cantor processes and Cantor sets,the above denseness conclusion in the third chapter are reviewed from the perspective of measure theory in the fourth chapter.For example,consider the case p=3,it is illustrated that {L([0,x])/(μC([0,x]))θ:x ∈C∩[2/3,1} is dense in[1,2],where L is the Lebesgue measure,and μC is the classical Cantor measure on Cantor ternary set C,and θ=log23.At the same time,we also construct a function Cp with period 1,and get an expression of an,that is an=nlogspCp(logsn).In the final chapter,we provide an overview of the work we have done and raise questions for further discussion.