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 $p \in \mathbb{N}$ , there is some $n \in \mathbb{N}$ such that $\{n, n+1, n+2, ... , n+p \} \subset T$ .

Examples

Trivially $\mathbb{N}$ 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:

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

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 $\mathbb{N}$ with the addition operation $+$ , this definition is equivalent to the one given above.

References

J. McLeod, "[http://topology.nipissingu.ca/tp/reprints/v25/tp25217.pdf Some Notions of Size in Partial Semigroups]", Topology Proceedings, Vol. 25 (Summer 2000), pp. 317-332.
Vitaly 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)
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.

Category:Semigroup theory

Category:Ergodic theory

Thick set

Examples

Generalisations

See also

References