Expansion Of Real Numbers And Complexity Of Sequences

Areal number may yield an infinite sequence via expansions with respect to a given base.In this way,real numbers in algebra and the sequences in combinatorics of words are linked together,and it's also a new exploration of the real number theory.On the basis of the previous study,we consider the expansions of the real number with respect to two multiplicatively dependent bases,and obtain some results on the complexity of expansion sequences.Chapter 1 mainly introduces the relevant background and significance of researching.Chapter 2 introduces the basic knowledge of real numbers'expansion with respect to a given base and combinatorics of words,mainly including the basic properties of word and related symbols,the function of complexity of the words,Sturmian words and their equivalent definitions,and the definition and related properties of quasiSturmian words,etc.Based on Chapters 1 and 2 and the references,this article mainly solves the followi?ng problems in Chapter 3:Firstly,if the b-ary expansion of an irrational real number ? is a Sturmian word on ?0,1?,then it's bk-ary expansion is not a Sturmian word,and exact complexity of the expansion is determined.Secondly,for a Sturmian word s,there exits homomorphic a mapping ? such that the bk-ary expansion of ?(s)is a Sturmian word.Thirdly,if the b-ary expansion of an irrational real number ? is a quasi-SturmiAn word on ?0,1?,the upper bound estimation of the complexity of the bk-ary expans-ion is given.Lastly,according to[1],let r,s?2 be multiplicatively dependent integers and m,l be the smallest positive integers such that rm=sl.Then every irrational real number satisfies(?)(p(n,?,r)+p{n,?,s)-2n?m+l This article describes the expansion sequence when the equal sign in the above inequality is taken.