Piecewise syndetic set

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 are arbitrarily long intervals of where the gaps in S are bounded by some constant b.

Properties[edit]

  • 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 Largeness[edit]

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

See also[edit]

Notes[edit]

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

References[edit]