Piecewise syndetic set

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

[math]\displaystyle{ x+F \subset \bigcup_{n \in G} (S-n) }[/math]

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