| Number theory is the theory of numbers, its object of study can include varioustypes of numbers. But the main object of study is integer, especially the positive inte-ger. Recent centuries, many mathematicians were committed to studying the propertyof positive integers and obtained some theoretical results. Thus they played a role inpromoting the development of the number theory and mathematics. The study of pos-itive integer is seemingly simple. Actually, its property is profound and complex andrelate to the nature and primitive of mathematics.Solutions of many problems in number theory need some information on size ofparticular sets of positive integers. The common point in all densities is that they arefunctions f defined on the power of N, such that f(?) = 0, f(N) = 1 and if A (?) Band f(A), f(B) are defined, then f(A)≤f(B). For specific issues, we need to definesome specific densities, asymptotic and logarithmic densities provide one of the mostimportant ways how to do it. Extending previous results, we give a new description ofthe density set, i.e. the set of all pairs of densities - upper and lower - of all subsets of agiven set of positive integers.In the introductory chapter, we first introduce the background of the study and thenwe review the known results and make some remarks. In Chapter 2, we give some relateddefinitions, notations, elementary properties and relevant conclusions. Then, we studythe relationship among several different densities, especially the asymptotic density andthe gap density. After that, we give three relevant propositions and prove them. Inthe last chapter we give the concepts of weighted density and density set. We thenprove that, when the weight function satisfies some conditions, the density set is convex.Further, we establish the joint Darboux property of the weighted density. Finally weprove that the density set is closed via an explicit characterization of its upper boundary. |
