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.)