Thick set

In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set formula_1, for every formula_2, there is some formula_3 such that formula_4.
Examples.
Trivially formula_5 is a thick set. Other well-known sets that are thick include non-primes and non-squares. Thick sets can also be sparse, for example:
formula_6
Generalisations.
The notion of a thick set can also be defined more generally for a semigroup, as follows. Given a semigroup formula_7 and formula_8, formula_9 is said to be "thick" if for any finite subset formula_10, there exists formula_11 such that
formula_12
It can be verified that when the semigroup under consideration is the natural numbers formula_5 with the addition operation formula_14, this definition is equivalent to the one given above.