Thick set

In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set $$T$$, for every $$p \in \mathbb{N}$$, there is some $$n \in \mathbb{N}$$ such that $$\{n, n+1, n+2, ..., n+p \} \subset T$$.

Examples
Trivially $$\mathbb{N}$$ 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:

$$\bigcup_{n \in \mathbb{N}} \{x:x=10^n +m:0\le m\le n\}.$$

Generalisations
The notion of a thick set can also be defined more generally for a semigroup, as follows. Given a semigroup $$(S, \cdot)$$ and $$A \subseteq S$$, $$A$$ is said to be thick if for any finite subset $$F \subseteq S$$, there exists $$x \in S$$ such that

$$F \cdot x = \{ f \cdot x : f \in F \} \subseteq A.$$

It can be verified that when the semigroup under consideration is the natural numbers $$\mathbb{N}$$ with the addition operation $$+$$, this definition is equivalent to the one given above.