The Study On Fatness And Thinness For Doubling Measures In Euclidean Space

In this dissertation, we studied the fatness and thinness for doubling measures in Euclidean space. The paper contains three parts.The first part, we studied that which compact subsets of the real line R only carries purely atomic doubling measures? and Which compact subsets of the real line R does not carry any purely atomic doubling measures? The above questions had come from Kaufman-Wu[50]. In that paper, they proved that all doubling measures on a compact metric space X are mutually absolutely continuous if and only if every doubling measure on X is purely atomic. Write X=Ex U Fx, where EX is the set of accumulation points of X and Fx is the set of isolated points. The main idea is to seek a suitable parameter to describe the relative size of Fx and Ex, we have the following results:Let X be compact in R such that Fx=X and Ex is perfect. If there is an element{Ek}∞k=0∈(?)X such that for any p>0, then every doubling measure on X is purely atomic. Let X be compact in R such that Fx=X and Ex is perfect. If there is an element{Ek}∞k=0∈(?)X such that for any r∈(0,1) then X does not carry any purely atomic doubling measures.The second part, concerns a special class of Moran sets, the A-Moran sets, and obtains the necessary and sufficient conditions to describe the thickness and thinness of λ-Moran sets. We have the following results:Let E=E({nk},{ck,i}) be a λ-Moran set. Then E is thick if and only if for all0<p<1. Let E=E({nk},{ck,i}) be a A-Moran set. Then E is thin if and only if for all p>1.The third part, we studied the fatness and thinness of the subsets of Rn. According to the size of sets in the sense of doubling measures, subsets of Rn can be divided into six classes. Sets in these six classes are respectively called very fat, fairly fat, minimally fat, very thin, fairly thin, and minimally thin. We mainly discuss that when does the product of n subsets of [0,1] belong to VF, FF, MF, VT, FT, or MT for doubling measures on [0.1]n? We have the following results:Let E1,…,En be arbitrary sets in [0,1], each of zero Lebesgue measure. Then E1×…×En∈VT if and only if Ei∈VT for some i; E1×…×En e MT if and only if Ei∈MT for any i. Let E1,…,En be arbitrary sets in [0,1], each of positive Lebesgue measure. Then if E1×…×En∈VF then Ei∈VF for any i; If Ei∈MF for some i then E1×…×En∈MF. The question, if the inverses of two propositions in the above Theorem are true for the product of general sets, is open, no a proof nor any counterexample. We shall prove that they are true for the product of uniform Cantor sets. Let E1,…,En be uniform Cantor sets in [0,1], each of positive Lebesgue measure. Then E1×…×En∈VF if and only if Ei∈VF for any i; E1×…×En∈MF if and only if Ei∈MF for some i.