Study On The Dimensions Of A Class Of Generalized Cantor Sets

Since the 19^th century,the understanding of fractal geometry has been transferred from natural phenomena to mathematical problems.Several typical fractal sets have been proposed and many studies have been made.Especially in the late 19^th century,fractal geometry has officially become an independent subject through in depth study of the nature of fractal sets.The middle third Cantor set is the most typical set in fractal geometry,and it is also the most easily constructed fractal.From the establishment of the fractal geometry discipline,many studies have been done on the generalized structure of the middle third Cantor set.In recent years,with the development of fractal geometry,many scholars have studied many generalized fractal sets on the basis of the middle third Cantor set,and obtained and proved a series of properties.Since the middle third Cantor set is one of the most classical types in fractal research,the basic nature of its structure has led many researchers to do a lot of work in this field.On the basis of the middle third Cantor set by previous studies,we concerned a class of generalized Cantor set,which the Cantor set is divided into 2k+1 equal parts,Furthermore,we discuss its characteristics and properties.By applying the principles of mass distribution,we obtain more sharp estimates of its lower bound,and we obtain its Hausdorff measure by using the finite cover lemma to compare the various basic interval.Moreover,we calculate box dimension and packing dimension of the generalized Cantor set.Finally,the relation and differences between the Hausdorff dimension,the box dimension and the Packing dimension in the computation of the dimension of the generalized Cantor set are given.