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Thick set

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

Examples

Trivially N {\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:

n N { x : x = 10 n + m : 0 m n } . {\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 ( S , ) {\displaystyle (S,\cdot )} and A S {\displaystyle A\subseteq S} , A {\displaystyle A} is said to be thick if for any finite subset F S {\displaystyle F\subseteq S} , there exists x S {\displaystyle x\in S} such that

F x = { f x : f F } A . {\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 N {\displaystyle \mathbb {N} } with the addition operation + {\displaystyle +} , this definition is equivalent to the one given above.

See also

References

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