Quasi-Assouad Dimension Of The Product Sets

Fractal geometry provides well ideas and methods for studying irregular sets.It is well known that fractal dimensions include hausdorff dimension,box dimension and packing dimension.At present,the research on the dimension of fractal sets focuses on the Assouad dimension and the quasi-Assouad dimension of fractal sets.In the Euclidean space spaces Rd,if E F(?)Rd,then dimHE + dimHF ?dimH(E×F)? dimHE + dimPF,(1)dimHE + dimPF ? dimP(E×F)? dimPE + dimPF,(2)dimBE + dimBF ? dimB(E×F)?dimBE + dimBF.(3)Where dimH,dimB,dimB,dimP denote the Hausdorff,upper box-counting,lower box-counting,packing.About the Assouad dimension of product sets,in[10],it prove that given 0??????????+?,where exist compact subsets E,F of the Euclidean space such that dim AE=?,dimAF = ?,dimA(Ex F)= ?.That is to say compact sets E,F satisfy:dimAE ?dimA(E×F)? dimAE + dimAF.In this paper,we study the quasi-Assouad dimension of products of fractal sets.We get similar results as[10]:we prove that given 0????,?????+?,where exist compact subsetsE,Fof the Euclidean space such that dimqAE=?,dimqAF=?,dimqA(E × F)= ?.That is to say compact sets E,F satisfy:dimqAE E?dimqA(E × F)? dimqAE + dimqAF.In[10],it proves their results by mainly using the uniform Cantor set to construct compact sets E,F.In this paper,we prove our conclusions by using digital limit sets to construct compact sets E,F.This is a difference between us and[10].In this paper,it introduces the development process of fractal geometry and dimensions of fractal in the first chapter;in the second chapter,it gives the basic concepts and theories of fractal dimensions;in the third chapter,it mainly introduces the existed results of the dimension of products of sets and the main results and proofs that we obtain.