Thick set

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, "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.