Hausdorff Dimension Of Homogeneous Complete Sets

In this paper,the concept of homogeneous complete set is introduced as an extension of Cantor set.The exact dimension is determined according to the length of the basic interval and the distance between intervals,and the following theorem is proposed and proved.Let E=E(J,{nk?,{ck?,{?k,j?)be a homogeneous complete set.Suppose there exist positive constants c1,c2,c3 such that for any k?1,Where J is the initial basic interval,nk is the number of basic intervals generated,ck is the length of the basic interval,and ?k,j is the distance between the intervals,at least one of the following four conditions is satisfied:(A)max1?l?nk-1?k,l?c1min1?l?nk-1?k,l;(B)max1?l?nk-1?k,l?c2·c1c2…ck;(C)nk·min1?l?nk-1?k,l?c3·c1c2…ck-1;(D)(?)Then(?)The first chapter is the introduction,which briefly introduces the research background and current situation of this article and the main theorems of the research.The second chapter preliminarily elaborates the basic knowledge needed to prove the main theorem in detail,including the construction of homogeneous complete sets,and the concepts and properties of Hausdorff dimension and measure.The third chapter is the proof of the main theorem,giving the proof of the main theorem involved in this article.The fourth chapter summarizes and ponders,summarizes the full text and puts forward more in-depth thoughts on the main theorems.