# Piecewise syndetic set

In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers.

A set ${\displaystyle S\subset {\mathbb {N} }}$ is called piecewise syndetic if there exists a finite subset G of ${\displaystyle \mathbb {N} }$ such that for every finite subset F of ${\displaystyle \mathbb {N} }$ there exists an ${\displaystyle x\in {\mathbb {N} }}$ such that

${\displaystyle x+F\subset \bigcup _{n\in G}(S-n)}$

where ${\displaystyle S-n=\{m\in {\mathbb {N} }:m+n\in S\}}$. Equivalently, S is piecewise syndetic if there are arbitrarily long intervals of ${\displaystyle \mathbb {N} }$ where the gaps in S are bounded by some constant b.

## Properties

• If S is piecewise syndetic then S contains arbitrarily long arithmetic progressions.

## Other Notions of Largeness

There are many alternative definitions of largeness that also usefully distinguish subsets of natural numbers: