Density

Let X be a finite set. The usual definition of the density of a subset Y of X is |Y|/|X|, that is, the size of Y divided by the size of X. In particular, if [math]\displaystyle{ \mathcal{A} }[/math] is a subset of [math]\displaystyle{ [3]^n }[/math] then its density is [math]\displaystyle{ 3^{-n}|\mathcal{A}|. }[/math]

One speaks loosely of a set [math]\displaystyle{ \mathcal{A}\subset[3]^n }[/math] being dense if its density [math]\displaystyle{ \delta }[/math] is bounded below by a positive constant that is independent of n. Strictly speaking, this definition applies to sequences of sets with n tending to infinity, but it is a very useful way of talking.

Sometimes it is helpful to consider other probability measures on [math]\displaystyle{ [3]^n, }[/math] such as equal-slices density. Then the words "density" and "dense" have obviously analogous uses.