Piecewise syndetic set

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

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

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

where $S-n=\{m\in \mathbb{N}:m+n\in S\}$. Equivalently, S is piecewise syndetic if there is a constant b such that there are arbitrarily long intervals of $\mathbb{N}$ 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 $\beta \mathbb{N}$, the Stone–Čech compactification of the natural numbers.
• Partition regularity: if $S$ is piecewise syndetic and $S=C_1\cup C_2\cup \dots \cup C_n$, then for some $i\le n$, $C_i$ contains a piecewise syndetic set. (Brown, 1968)
• If A and B are subsets of $\mathbb{N}$ with positive upper Banach density, then $A+B=\{a+b:a\in A,b\in B\}$ is piecewise syndetic.

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

References

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