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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.

Contents

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

x + F ⊂ ⋃ n ∈ G ( S − n )

where S − n = { m ∈ N : m + n ∈ S } . Equivalently, S is piecewise syndetic if there are arbitrarily long intervals of N where the gaps in S are bounded by some constant 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 , the Stone–Čech compactification of the natural numbers.
  • Partition regularity: if S is piecewise syndetic and S = C 1 ∪ C 2 ∪ . . . ∪ C n , then for some i ≤ n , C i contains a piecewise syndetic set. (Brown, 1968)
  • If A and B are subsets of N , and A and B have positive upper Banach density, then A + B = { a + b : a ∈ A , b ∈ B } is piecewise syndetic

    • Other notions of largeness

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

  • Cofiniteness
  • IP set
  • member of a nonprincipal ultrafilter
  • positive upper density
  • syndetic set
  • thick set

    • References

    Piecewise syndetic set Wikipedia
    (Text) CC BY-SA


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