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The density of a topological space is the least ... An alternative definition of dense set in the case of metric spaces is the following.
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. [3]
The Schnirelmann density of a set of natural numbers A is defined as = (), where A(n) denotes the number of elements of A not exceeding n and inf is infimum. [3] The Schnirelmann density is well-defined even if the limit of A(n)/n as n → ∞ fails to exist (see upper and lower asymptotic density).
The set of points in the plane at which the density is neither 0 nor 1 is non-empty (the square boundary), but it is negligible. The Lebesgue density theorem is a particular case of the Lebesgue differentiation theorem. Thus, this theorem is also true for every finite Borel measure on R n instead of Lebesgue measure, see Discussion.
The logarithmic density of a set of integers from 1 to can be defined by setting the weight of each integer to be /, and dividing the total weight of the set by the th partial sum of the harmonic series (or, equivalently for the purposes of asymptotic analysis, dividing by ). The resulting number is 1 or close to 1 when the set includes all ...
For example, if A is the set of all primes, it is the Riemann zeta function which has a pole of order 1 at s = 1, so the set of all primes has Dirichlet density 1. More generally, one can define the Dirichlet density of a sequence of primes (or prime powers), possibly with repetitions, in the same way.
Construction of a two-dimensional Danzer set with growth rate () from overlaid rectangular grids of aspect ratio 1:1, 1:9, 1:81, etc. In geometry, a Danzer set is a set of points that touches every convex body of unit volume. Ludwig Danzer asked whether it is possible for such a set to have bounded density.
Dense set — a subset of a topological space whose closure is the whole space; Dense-in-itself — a subset of a topological space such that does not contain an isolated point; Kripke semantics — a dense accessibility relation corresponds to the axiom