Asymptotic Behavior Of The Maximal Run-length Function In The Lüroth Expansion

In this dissertation,we study the asymptotic behavior of the maximal run-length function in the Lüroth expansion.Erdos and Renyi[1]obtained a large number law for Zn dyadic run-length func-tion in 1970.Then Erdos and Revesz[2]estimated the rate of asymptotic growth of Zn accurately,which can be generalized to n-adic expansion naturally.N-adic expansion is a finite symbolic system.We estimate the growth speed of the maximal run-length function in Lüroth system that is an infinite symbolic system precisely.This thesis is divided into three chapters.In the first two chapters,the relevant research background and preliminaries are presented.In the third chapter that is the main part of this thesis,we study the growth rate of maximal run-length function in the Liiroth system.We need to introduce some notation so as to state the result.For every x?(0,1],we denote the Lüroth expansion of x by[a1(x),a2(x),…,ak(x),…].We callln(x)maxj?2{k:ai+1(x)= … ai+k(x)=j,for some 0 ? i ?n-k}the maximal run-length function of x,which represents the longest run of the same symbol in the first n digits of x.[z]denotes the maximal integer that is less than or equal to z real number and e is the base of natural logarithm.Let ? be any positive number.We prove the pri-mary result in this thesis as follows.(1)For almost all x E(0,1],there exists a finite N = N(x,?)such that ln(x)?[log2n-log2 log2 log2 n + log2 log2 e-2-?]if n?N.(2)For almost all x ?(0,1],there exists an infinite sequence Ni=Ni(x,?)(i=1,2,…)of integers such thatlNi(x)<[log2 Ni-log2 log2 log2 Ni + log2 log2 e-1 + ?].