Syndetic set

In mathematics, a syndetic set is a subset of the natural numbers, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded.

Definition

A set $S \sub \mathbb{N}$ is called syndetic if for some finite subset F of $\mathbb{N}$

$\bigcup_{n \in F} (S-n) = \mathbb{N}$

where $S-n = \{m \in \mathbb{N} : m+n \in S \}$. Thus syndetic sets have "bounded gaps"; for a syndetic set $S$, there is an integer $p=p(S)$ such that $[a, a+1, a+2, ... , a+p] \bigcap S \neq \emptyset$ for any $a \in \mathbb{N}$.

See also

• Ergodic Ramsey theory
• Piecewise syndetic set
• Thick set

References

• J. McLeod, "Some Notions of Size in Partial Semigroups", Topology Proceedings, Vol. 25 (2000), pp. 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), pp. 18-36