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 $[a,a+1,a+2,\mathrm{...},a+p]\bigcap S\ne \mathrm{\varnothing }$ for any $a\in \mathbb{N}$.

See also

• Ergodic Ramsey theory
• Piecewise syndetic set
• Thick set