# Thick set

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

## Examples

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

${\displaystyle \bigcup _{n\in \mathbb {N} }\{x:x=10^{n}+m:0\leq m\leq n\}.}$

## Generalisations

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

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

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