Syndetic set

Short description: Type of subset of the natural numbers

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

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

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

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. ISBN 978-0-521-53365-2. http://www.math.ohio-state.edu/~vitaly/vbkatsiveli20march03.pdf. 
• 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.