Some Studies On The Normality Of The Expansions Of Numbers

Number theory and ergodic theory are two important fields in mathematics.In recent years,the theory of dynamical system has been applied more and more widely to the prob-lems of number theory.For example,we can provide a simple proof of the famous Borel normal number theorem by using the Brikhoff ergodic theorem.This paper deals with the normality of the expansions of numbers.We will explore the normality of numbers from the perspective of automaton and discuss selection rules realized by an automaton over{0,1}.We give three characterizations of the selection rules preserving normality.This paper is divided into six chapters.In the first chapter,we mainly introduce the relevant background knowledge and research status,such as dynamical system,automaton and the normality of expansions of numbers.In the second chapter we introduce several common expansions of numbers and their corresponding definitions of normality and give the basic concepts of automaton and selection rules and the connection between them.We also give definitions of the concepts of“conditionally deterministic”and“weakly renewal”.In the following three chapters we give several characterizations of selection rules preserving normality.In chapter 3,we prove that for a normal number??{0,1}^Nand a countable au-tomaton M,let(?,?)?{0,1}^N×?^Nbe the input-output sequence.Assume that there exists a mapping?from?to a finite set K such that(?,?)?{0,1}^N×K^N,where?(i)=?(?(i))(?i?N),is conditionally deterministic.Assume further that there ex-ists L?K such that?^-1L=F.Then,?[S(M,?)]is a normal number if S(M,?)has a positive lower density.In chapter 4,we present a sufficient condition for the selection rule to preserve normal-ity.We say that M^?is weakly renewal with the renewal word??{0,1}^lwhere l?1 if there exists a finite partition{?₁,···,?_k}of?such that for any i=1,···,k,there exists?_i??such that?(?,?)=?_ifor any???_i.A countable automaton M=(?,?,?₀,F)preserves normality if M^?is weakly renewal.In chapter 5,we consider the case when the set of states?of automaton M=(?,?,?₀,F)is a compact and metrizable set.Let?be a binary normal number and M=(?,?,?₀,F)be an automaton,where?is a compact metrizable set,?:?×{0,1}??,?₀??,F??.Let X be a countable dense subset of?.1.If X is open,let X^*=X?{?}be the one-point compactification of X.Suppose H(?)<?(see(5.5))under P(see(5.2)).Let?be defined as(5.1).If?is con-tinuous,there exists L(?)X^*such that?^-1L=F and?_?^U(L)>0 for any?_?^U,then ?[S(M,?)]is a normal number.2.If X is closed,then?is countable.Suppose H(?₀)<?(see(5.9))under?(see(5.6)).If?:?×{0,1}??is continuous and S(?,F)has a positive lower density,then?[S(M,?)]is a normal number.In the last chapter,we summarize the main results and put forward the questions to be considered in the future.