The upper Banach density of a set 
of integers is
 |
In the ergodic theory approach to Szemerédi's theorem, upper 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.
For a countable discrete Abelian group 
,
the definition extends using Følner sequences.
If 
is such a sequence, put
 |
The upper Banach density is 
. The multiplicative upper Banach
density 
on the positive integers is obtained by using
multiplication instead of addition.
High upper Banach density does not force every natural infinite configuration. For every 
,
Ackelsberg (2026) constructs a set 
of positive integers with
that contains no set 
for any infinite 
and 
. He also proves that if 
has infinite index in 
, then 
has sets of upper Banach density
arbitrarily close to 1 containing no shifted restricted sumset 
for
any infinite 
.
In the integers, an analogous high-density counterexample
excludes every configuration 
.
Problem." Electron. J. Combin. 33, P3.89, 2026.