Piecewise syndetic set

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

A set $S \sub \mathbb{N}$ 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 are arbitrarily long intervals of $\mathbb{N}$ where the gaps in S are bounded by some constant 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 $S = C_1 \cup C_2 \cup ... \cup C_n$, then for some $i \leq n$, $C_i$ contains a piecewise syndetic set. (Brown, 1968)

• If A and B are subsets of $\mathbb{N}$, and A and B have 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

• J. McLeod, "Some Notions of Size in Partial Semigroups" Topology Proceedings 25 (2000), 317-332
• Vitaly Bergelson, "Minimal Idempotents and Ergodic Ramsey Theory", Topics in Dynamics and Ergodic Theory 8-39, London Math. Soc. Lecture Note Series 310, Cambridge Univ. Press, Cambridge, (2003)
• Vitaly Bergelson, N. Hindman, "Partition regular structures contained in large sets are abundant", J. Comb. Theory (Series A) 93 (2001), 18-36
• T. Brown, "An interesting combinatorial method in the theory of locally finite semigroups", Pacific J. Math. 36, no. 2 (1971), 285–289.