Thick set
In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set [math]\displaystyle{ T }[/math], for every [math]\displaystyle{ p \in \mathbb{N} }[/math], there is some [math]\displaystyle{ n \in \mathbb{N} }[/math] such that [math]\displaystyle{ \{n, n+1, n+2, ... , n+p \} \subset T }[/math].
Examples
Trivially [math]\displaystyle{ \mathbb{N} }[/math] 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:
[math]\displaystyle{ \bigcup_{n \in \mathbb{N}} \{x:x=10^n +m:0\le m\le n\}. }[/math]
Generalisations
The notion of a thick set can also be defined more generally for a semigroup, as follows. Given a semigroup [math]\displaystyle{ (S, \cdot) }[/math] and [math]\displaystyle{ A \subseteq S }[/math], [math]\displaystyle{ A }[/math] is said to be thick if for any finite subset [math]\displaystyle{ F \subseteq S }[/math], there exists [math]\displaystyle{ x \in S }[/math] such that
[math]\displaystyle{ F \cdot x = \{ f \cdot x : f \in F \} \subseteq A. }[/math]
It can be verified that when the semigroup under consideration is the natural numbers [math]\displaystyle{ \mathbb{N} }[/math] with the addition operation [math]\displaystyle{ + }[/math], this definition is equivalent to the one given above.
See also
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.
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