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 [1, n], then the probability that it belongs to A is the ratio of the number of elements of A in [1, n] to the total number of elements in [1, n]. 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.
- a(n)/n → α as n → +∞.
It follows from the definition that if a set A has natural density α then 0 ≤ α ≤ 1.
Upper and lower asymptotic density
Let be a subset of the set of natural numbers For any put and .
Define the upper asymptotic density of by
where lim sup is the limit superior. is also known simply as the upper density of
Similarly, , the lower asymptotic density of , is defined by
One may say has asymptotic density if , in which case is equal to this common value.
This definition can be restated in the following way:
if the limit exists.
It can be proven that the definitions imply that the following also holds. If one were to write a subset of as an increasing sequence
and if the limit exists.
A somewhat weaker notion of density is the upper Banach density; given a set , define as
Properties and examples
- If d(A) exists for some set A, then for the complement set we have d(Ac) = 1 − d(A).
- If , , and exist, then .
- The density d(N) of the entire set of natural numbers is equal to 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 . More generally, the set of all n-th-power-free numbers for any natural n has density , where is the Riemann zeta function.
- The set of abundant numbers has non-zero density. Marc Deléglise showed in 1998 that the density of the set of abundant numbers and perfect numbers is 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
- whereas its lower density is
- The set of numbers whose decimal expansion begins with the digit 1 similarly has no natural density: the lower density is 1/9 and the upper density is 5/9.
- Consider an equidistributed sequence in and define a monotone family of sets :
- Then, by definition, for all .
- If S is a set of positive upper density then Szemerédi's theorem states that S contains arbitrarily large finite arithmetic progressions, and the Furstenberg–Sárközy theorem states that some two members of S differ by a square number.
Other density functions
Other density functions on subsets of the natural numbers may be defined analogously. For example, the logarithmic density of a set A is defined as the limit (if it exists)
Upper and lower logarithmic densities are defined analogously as well.
- Tenenbaum (1995) p.261
- Nathanson (2000) pp.256–257
- Hall, Richard R.; Tenenbaum, Gérald (1988). Divisors. Cambridge Tracts in Mathematics. 90. Cambridge: Cambridge University Press. p. 95. ISBN 0-521-34056-X. Zbl 0653.10001.
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- Nathanson, Melvyn B. (2000). Elementary Methods in Number Theory. Graduate Texts in Mathematics. 195. Springer-Verlag. ISBN 0387989129. Zbl 0953.11002.
- Niven, Ivan (1951). "The asymptotic density of sequences". Bulletin of the American Mathematical Society. 57: 420–434. doi:10.1090/s0002-9904-1951-09543-9. MR 0044561. Zbl 0044.03603.
- Steuding, Jörn (2002). "Probabilistic number theory" (PDF). Archived from the original (PDF) on December 22, 2011. Retrieved 2014-11-16.
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