Piecewise syndetic set

In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers. A set $S\subset \mathbb{\mathbb{N}}$ is called piecewise syndetic if there exists a finite subset G of $\mathbb{\mathbb{N}}$ such that for every finite subset F of $\mathbb{\mathbb{N}}$ there exists an $x\in \mathbb{\mathbb{N}}$ such that

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

where $S-n=\{m\in \mathbb{\mathbb{N}}:m+n\in S\}$. Equivalently, S is piecewise syndetic if there is a constant b such that there are arbitrarily long intervals of $\mathbb{\mathbb{N}}$ 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 $\beta \mathbb{\mathbb{N}}$, 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 $\mathbb{\mathbb{N}}$ 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

• McLeod, Jillian (2000). "Some Notions of Size in Partial Semigroups". Topology Proceedings 25 (Summer 2000): 317—332. http://topology.nipissingu.ca/tp/reprints/v25/tp25217.pdf. 
• "Minimal Idempotents and Ergodic Ramsey Theory". Topics in Dynamics and Ergodic Theory. London Mathematical Society Lecture Note Series. 310. Cambridge University Press, Cambridge. 2003. pp. 8—39. doi:10.1017/CBO9780511546716.004. http://www.math.ohio-state.edu/~vitaly/vbkatsiveli20march03.pdf. 
• "Partition regular structures contained in large sets are abundant". Journal of Combinatorial Theory. Series A 93 (1): 18—36. 2001. doi:10.1006/jcta.2000.3061. 
• Brown, Thomas Craig (1971). "An interesting combinatorial method in the theory of locally finite semigroups". Pacific Journal of Mathematics 36 (2): 285—289. doi:10.2140/pjm.1971.36.285. http://projecteuclid.org/euclid.pjm/1102971066.