Banach Density
The Banach density of a set A of integers is defined as
![画像: lim_(d->infty)max_(n)(|{A intersection [n+1,...,n+d]}|)/d,](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/BanachDensity/NumberedEquation1.svg){kind=link}
if the limit exists. If the lim is replaced with limsup or liminf, 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.)
See also
Szemerédi's TheoremThis entry contributed by Kevin O'Bryant
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O'Bryant, Kevin. "Banach Density." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BanachDensity.html