Properties Of The Length Of Basic Interval In Beta Expansions

The research of the real number started from long time ago. We contacted from thebeginning of the decimal expansion to binary expansion and pushed to b-ary expansion.Until1957, A. Renyi had introduced beta-expansion of real number, which is consideredto be the extension of b-ary expansion of real number. Then Parry systematically studiedthe beta-expansion, such as the characterization admissible sequence and invariant mea-sure, and so on. So this paper research that let β>1be a real number, if we write (?)(x)for the n-th order basic interval containing x and|(?)(x)|for the length of (?)(x), thenwe can obtain the exact value of the length of (?)(x).The thesis is organized as follows: in the introductory chapter we mainly introducethe background and current situation of the research, and then review some previous workand the conclusion.In the second chapter, we cite some definitions, elementary properties and relevantconclusions.In the third chapter, we consider the invariant measure under the beta-transformation.We can find a unique normalised measure equivalent to Lebesgue measure.In the fourth chapter, We define the n-th order basic interval and full basic intervals,and we establish the full basic interval. Then we cite the digit sequence of1and obtainup and low bound of the length of basic interval. So we obtain a relationship between thelength of basic interval and the beta-expansion of1, which enables us to calculate thevalue of the length of basic interval.