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

• 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