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 is a constant b such that there are arbitrarily long intervals of ${\displaystyle \mathbb {N} }$ where the gaps in S are bounded by b.

Properties

• A set is piecewise syndetic if and only if it is the intersection of a syndetic set and a thick set.
• If S is piecewise syndetic then S contains arbitrarily long arithmetic progressions.
• A set S is piecewise syndetic if and only if there exists some ultrafilter U which contains S and U is in the smallest two-sided ideal of ${\displaystyle \beta \mathbb {N} }$, the Stone–Čech compactification of the natural numbers.
• Partition regularity: if ${\displaystyle S}$ is piecewise syndetic and ${\displaystyle S=C_{1}\cup C_{2}\cup \dots \cup C_{n}}$, then for some ${\displaystyle i\leq n}$, ${\displaystyle C_{i}}$ contains a piecewise syndetic set. (Brown, 1968)
• If A and B are subsets of ${\displaystyle \mathbb {N} }$ with positive upper Banach density, then ${\displaystyle A+B=\{a+b:a\in A,\,b\in B\}}$ is piecewise syndetic.^[1]

Other notions of largeness

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

• Cofiniteness
• IP set
• member of a nonprincipal ultrafilter
• positive upper density
• syndetic set
• thick set

See also

• Ergodic Ramsey theory