Assouad Dimension Of A Class Of Generalized Set Defined By Digit Restrictions

In recent years,more and more researches have been done on the Assouad dimension of sets.The Assouad dimension of some special sets have been studied and calculated by many scholars both at home and abroad.In this paper,we studied a kind of generalized set defined by digit restrictions and calculated its Assouad dimension.The conclusion is as follows:Let Es,（?）is a set defined by digit restrictions and set A.S is a subset of positive integer set N.For any integer m in set N.Set Wm={ω₁ω₂…ω_m:ωk∈{1,2} if k∈S,ω_k=1if k（?）S,1≤k≤m}.If word ω=ω1ω2…ωn and v=v1v2…vm,then ω*v=ω1…ωnv1…vm.Define set（?）and the closed subset A={I_ω:ω∈W}:（?）I=[0,1]satisfies the following two relations:（1）For any word ω in set Wm,set I_ω is geometrically similar to set I,that is,there is a similarity mapping fω:R→R such that I_ω=fω（I）and |I_ω|=cω|I|.Without loss of generality,set Iφ=I.If word ω=ω1ω2…ωn,then the compression radio cω=cω1cω2…cωn.（2）For any integer m≥0 and any word ω in set Wm,when integer m+1∈S,the subsets I_ω*1 and I_ω*2 of set I_ω are disjoint and |I_ω*1|=ci|I_ω|,i=1,2;when integer m+1（?）S,we only take the subset I_ω*1 and |I_ω*1|=c1|I_ω|.Let C₁ and C₂ be the numbers given when the set ES.A is defined,and they are between 0 and 1,let s*be the only real number that satisfies the following equation（?）,where（?）is the upper Banach density of set S,then dim_AE_S,A=s*.