Thick set

Short description: Set of integers containing arbitrarily long intervals

In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set $T$, for every $p\in \mathbb{\mathbb{N}}$, there is some $n\in \mathbb{\mathbb{N}}$ such that $\{n,n+1,n+2,...,n+p\}\subset T$.

Examples

Trivially $\mathbb{\mathbb{N}}$ is a thick set. Other well-known sets that are thick include non-primes and non-squares. Thick sets can also be sparse, for example:

$$\bigcup _{n\in \mathbb{\mathbb{N}}}\{x:x=10^n+m:0\le m\le n\}.$$

Generalisations

The notion of a thick set can also be defined more generally for a semigroup, as follows. Given a semigroup $(S,\cdot )$ and $A\subseteq S$, $A$ is said to be thick if for any finite subset $F\subseteq S$, there exists $x\in S$ such that

$$F\cdot x=\{f\cdot x:f\in F\}\subseteq A.$$

It can be verified that when the semigroup under consideration is the natural numbers $\mathbb{\mathbb{N}}$ with the addition operation $+$, this definition is equivalent to the one given above.

See also

• Cofinal (mathematics)
• Cofiniteness
• Ergodic Ramsey theory
• Piecewise syndetic set
• Syndetic set

References

• J. McLeod, "Some Notions of Size in Partial Semigroups", Topology Proceedings, Vol. 25 (Summer 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", Journal of Combinatorial Theory, Series A 93 (2001), pp. 18-36
• N. Hindman, D. Strauss. Algebra in the Stone-Čech Compactification. p104, Def. 4.45.