# Syndetic set

In mathematics, a syndetic set is a subset of the natural numbers, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded.

## Definition

A set ${\displaystyle S\subset \mathbb {N} }$ is called syndetic if for some finite subset F of ${\displaystyle \mathbb {N} }$

${\displaystyle \bigcup _{n\in F}(S-n)=\mathbb {N} }$

where ${\displaystyle S-n=\{m\in \mathbb {N} :m+n\in S\}}$. Thus syndetic sets have "bounded gaps"; for a syndetic set ${\displaystyle S}$, there is an integer ${\displaystyle p=p(S)}$ such that ${\displaystyle [a,a+1,a+2,...,a+p]\bigcap S\neq \emptyset }$ for any ${\displaystyle a\in \mathbb {N} }$.