?-expansion And Continued Fraction Expansion

In this thesis,we further study the decisive relationship of beta-expansion and continued fraction expansion.The thesis is organized as follows:In the first chapter,we summarize the research status of the determinant relationship between two kinds of real numbers,and point out the research content of this paper on the basis of previous research work.In the second chapter,we give the related knowledges about continued fraction expansion and beta-expansion.In the third chapter,based on the definitions of n-th order basic intervals and full intervals in beta-expansion,the distribution properties of basic intervals are discussed.In the chapter four,the decisive relationship between the beta-expansion and the continued fraction expansion of irrational numbers is introduced,then the intermediate value theorem is obtained,and an example is given to verify the decisive relationship at a particular point.Next,the approximation degree to an irrational number by beta expansi-on and continued fraction expansion is introduced.When log?>6log2/?2,the approxim-ation degree to an irrational by beta-expansion is better than that by continued fraction,when log?<6log2/?2,the approximation degree to an irrational by continued fraction is better than that by beta-expansion.