Thick set

In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set , for every , there is some such that .


Trivially   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   and  ,   is said to be thick if for any finite subset  , there exists   such that


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.

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