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

Let $mathcal{P}_f(mathbb{N})$ denote the set of finite subsets of $mathbb{N}$ . Then a set $S sub mathbb{N}$ is called syndetic if for some $F in mathcal{P}_f(mathbb{N})$

: $igcup_{n in F} (S-n) = mathbb{N}$

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 $[a, a+1, a+2, ... , a+p] igcap S
eq emptyset$ for any $a in mathbb{N}$ .

ee also

*Piecewise syndetic set
*Thick set
*Ergodic Ramsey theory

References

* J. McLeod, " [http://www.mtholyoke.edu/%7Ejmcleod/somenotionsofsize.pdf Some Notions of Size in Partial Semigroups] ", "Topology Proceedings, Vol. 25 (2000), pp. 317-332
* V. Bergelson, " [http://www.math.ohio-state.edu/~vitaly/vbkatsiveli20march03.pdf 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)
* V. Bergelson, N. Hindman, " [http://members.aol.com/nhfiles2/pdf/large.pdf Partition regular structures contained in large sets are abundant] ", "J. Comb. Theory (Series A)" 93 (2001), pp. 18-36

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