Thick set
# 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:

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

## Generalisations

The notion of a thick set can also be defined more generally for a semigroup, as follows. Given a semigroup $(S,\cdot )$ 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.

