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 $S\subset \mathbb {N}$ 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

• McLeod, Jillian (2000). "Some Notions of Size in Partial Semigroups" (PDF). Topology Proceedings. 25 (Summer 2000): 317–332.
• Bergelson, Vitaly (2003). "Minimal Idempotents and Ergodic Ramsey Theory" (PDF). Topics in Dynamics and Ergodic Theory. London Mathematical Society Lecture Note Series. Vol. 310. Cambridge University Press, Cambridge. pp. 8–39. doi:10.1017/CBO9780511546716.004.
• Bergelson, Vitaly; Hindman, Neil (2001). "Partition regular structures contained in large sets are abundant". Journal of Combinatorial Theory. Series A. 93 (1): 18–36. doi:10.1006/jcta.2000.3061.