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

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

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

