Modified Intermediate Dimensions

Falconer et al.introduced the concept of upper and lower intermediate dimensions,but the upper and lower intermediate dimensions don’t possess countable stability.Motivated by the idea that modified upper box dimension is countably stable,in this paper we modify the upper and lower intermediate dimensions and obtain the modified upper and lower intermediate dimensions.we find the modified upper and lower intermediate measures to induce the modified upper and lower intermediate dimensions respectively.Because the relationships between the upper and lower intermediate dimension and the packing dimension are unknown,in this paper gives the concept of the upper and lower intermediate packing dimensions and the relationships between the upper and lower intermediate dimensions,Hausdorff dimension,upper and lower intermediate packing dimensions and packing dimension are also studied.Moreover,this paper systematically discusses some basic properties of the modified upper and lower intermediate dimensions,the upper and lower intermediate packing dimensions.The structure of this paper is as follows: The first chapter introduces the relevant research background and some results of this topic,and expounds the research content and purpose of this paper.Chapter 2 gives the basic preparation knowledge related to this paper,including the definitions of Hausdorff measure and dimension;the definitions of packing measure and dimension;the definitions of box dimension;the definitions of modified upper and lower intermediate measure and dimension;and the definitions of upper and lower intermediate packing measure and dimension and the properties of the modified upper and lower intermediate dimensions and the upper and lower intermediate packing dimensions.The third chapter is devoted to the proof of the main theorem.Finally,we present some problems related to this research.