Piecewise syndetic set

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

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

where . Equivalently, S is piecewise syndetic if there is a constant b such that there are arbitrarily long intervals of where the gaps in S are bounded by b.

PropertiesEdit

  • 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  , the Stone–Čech compactification of the natural numbers.
  • Partition regularity: if   is piecewise syndetic and  , then for some  ,   contains a piecewise syndetic set. (Brown, 1968)
  • If A and B are subsets of  , and A and B have positive upper Banach density, then   is piecewise syndetic[1]

Other notions of largenessEdit

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

See alsoEdit

NotesEdit

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

ReferencesEdit