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Piecewise syndetic set

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In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers.

A set S ⊂ N {\displaystyle S\subset \mathbb {N} } is called piecewise syndetic if there exists a finite subset G of N {\displaystyle \mathbb {N} } such that for every finite subset F of N {\displaystyle \mathbb {N} } there exists an x ∈ N {\displaystyle x\in \mathbb {N} } such that

x + F ⊂ ⋃ n ∈ G ( S − n ) {\displaystyle x+F\subset \bigcup _{n\in G}(S-n)}

where S − n = { m ∈ N : m + n ∈ S } {\displaystyle S-n=\{m\in \mathbb {N} :m+n\in S\}} . Equivalently, S is piecewise syndetic if there is a constant b such that there are arbitrarily long intervals of N {\displaystyle \mathbb {N} } where the gaps in S are bounded by b.

Properties

  • A set is piecewise syndetic if and only if it is the intersection of a syndetic set and a thick set.
  • If S is piecewise syndetic then S contains arbitrarily long arithmetic progressions.
  • A set S is piecewise syndetic if and only if there exists some ultrafilter U which contains S and U is in the smallest two-sided ideal of β N {\displaystyle \beta \mathbb {N} } , the Stone–Čech compactification of the natural numbers.
  • Partition regularity: if S {\displaystyle S} is piecewise syndetic and S = C 1 ∪ C 2 ∪ ⋯ ∪ C n {\displaystyle S=C_{1}\cup C_{2}\cup \dots \cup C_{n}} , then for some i ≤ n {\displaystyle i\leq n} , C i {\displaystyle C_{i}} contains a piecewise syndetic set. (Brown, 1968)
  • If A and B are subsets of N {\displaystyle \mathbb {N} } with positive upper Banach density, then A + B = { a + b : a ∈ A , b ∈ B } {\displaystyle A+B=\{a+b:a\in A,\,b\in B\}} is piecewise syndetic.

Other notions of largeness

There are many alternative definitions of largeness that also usefully distinguish subsets of natural numbers:

See also

Notes

  1. R. Jin, Nonstandard Methods For Upper Banach Density Problems, Journal of Number Theory 91, (2001), 20-38.

References

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