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.
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 , 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 with positive upper Banach density, then is piecewise syndetic.R. Jin, Nonstandard Methods For Upper Banach Density Problems, Journal of Number Theory 91, (2001), 20-38.
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
See also Ergodic Ramsey theory
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