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 [math]\displaystyle{ S \sub \mathbb{N} }[/math] is called syndetic if for some finite subset [math]\displaystyle{ F }[/math] of [math]\displaystyle{ \mathbb{N} }[/math]

[math]\displaystyle{ \bigcup_{n \in F} (S-n) = \mathbb{N} }[/math]

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

See also

• Ergodic Ramsey theory
• Piecewise syndetic set
• Thick set

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.