In mathematics, a thick set is a set of integers that contains arbitrarily long intervals.
That is, given a thick set
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:
The notion of a thick set can also be defined more generally for a semigroup, as follows.
Given a semigroup
is said to be thick if for any finite subset
It can be verified that when the semigroup under consideration is the natural numbers
with the addition operation
, this definition is equivalent to the one given above.