Tree alignment

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

• 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