Thick set

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

Examples

Trivially ${\displaystyle \mathbb {N} }$ 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:

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

Generalisations

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

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

It can be verified that when the semigroup under consideration is the natural numbers ${\displaystyle \mathbb {N} }$ with the addition operation ${\displaystyle +}$, 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.