Banach Density

The Banach density of a set A of integers is defined as

 lim_(d->infty)max_(n)(|{A intersection [n+1,...,n+d]}|)/d,

if the limit exists. If the lim is replaced with lim sup or lim inf, 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 Theorem

This entry contributed by Kevin O'Bryant

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O'Bryant, Kevin. "Banach Density." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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