Natural density

In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one of the possibilities to measure how large a subset of the set of natural numbers is. In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one of the possibilities to measure how large a subset of the set of natural numbers is. Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. The notion of natural density makes this intuition precise. If an integer is randomly selected from the interval , then the probability that it belongs to A is the ratio of the number of elements of A in to the total number of elements in . If this probability tends to some limit as n tends to infinity, then this limit is referred to as the asymptotic density of A. This notion can be understood as a kind of probability of choosing a number from the set A. Indeed, the asymptotic density (as well as some other types of densities) is studied in probabilistic number theory. Asymptotic density contrasts, for example, with the Schnirelmann density. One drawback of asymptotic density is that it is not defined for all subsets of N {displaystyle mathbb {N} } . A subset A of positive integers has natural density α if the proportion of elements of A among all natural numbers from 1 to n converges to α as n tends to infinity. More explicitly, if one defines for any natural number n the counting function a(n) as the number of elements of A less than or equal to n, then the natural density of A being α exactly means that It follows from the definition that if a set A has natural density α then 0 ≤ α ≤ 1. Let A {displaystyle A} be a subset of the set of natural numbers N = { 1 , 2 , … } . {displaystyle mathbb {N} ={1,2,ldots }.} For any n ∈ N {displaystyle nin mathbb {N} } put A ( n ) = { 1 , 2 , … , n } ∩ A . {displaystyle A(n)={1,2,ldots ,n}cap A.} and a ( n ) = | A ( n ) | {displaystyle a(n)=|A(n)|} . Define the upper asymptotic density d ¯ ( A ) {displaystyle {overline {d}}(A)} of A {displaystyle A} by