The Banach density of a set 
of integers is defined as
![lim_(d->infty)max_(n)(|{A intersection [n+1,...,n+d]}|)/d,](/images/equations/BanachDensity/NumberedEquation1.svg) |
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.)
