Piecewise syndetic set

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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

  • If S is piecewise syndetic then S contains arbitrarily long arithmetic progressions.

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</math>.

References