Banach Density

The Banach density of a set  of integers is defined as

if the limit exists. If the  is replaced with  or , then the result is known as the upper or lower Banach density, respectively.

In the ergodic theory approach to Szemerédi's theorem, Banach density must be used. (Although the statements of Szemerédi's theorem with different types of density are equivalent, the proofs are not easily converted from one density type to the other.)