Natural Density - Examples

Examples

  • If d(A) exists for some set A, then for the complement set we have d(Ac) = 1 − d(A).
  • Obviously, d(N) = 1.
  • For any finite set F of positive integers, d(F) = 0.
  • If is the set of all squares, then d(A) = 0.
  • If is the set of all even numbers, then d(A) = 0.5 . Similarly, for any arithmetical progression we get d(A) = 1/a.
  • For the set P of all primes we get from the prime number theorem d(P) = 0.
  • The set of all square-free integers has density
  • The density of the set of abundant numbers is known to be between 0.2474 and 0.2480.
  • The set of numbers whose binary expansion contains an odd number of digits is an example of a set which does not have an asymptotic density, since the upper density of this set is
\overline d(A)=\lim_{m \rightarrow \infty} \frac{1+2^2+\cdots +2^{2m}}{2^{2m+1}-1} = \lim_{m \rightarrow \infty} \frac{2^{2m+2}-1}{3(2^{2m+1}-1)} = \frac 23\, ,
whereas its lower density is
\underline d(A)=\lim_{m \rightarrow \infty} \frac{1+2^2+\cdots +2^{2m}}{2^{2m+2}-1} = \lim_{m \rightarrow \infty} \frac{2^{2m+2}-1}{3(2^{2m+2}-1)} = \frac 13\, .
  • Consider an equidistributed sequence in and define a monotone family of sets :
Then, by definition, for all .

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