Piecewise syndetic set
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In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers.
Let 
denote the set of finite subsets of 
. Then a set 
is called piecewise syndetic if there exists 
such that for every 
there exists an 
such that
where 
. Informally, S is piecewise syndetic if there is some fixed bound b and S contains arbitrarily long intervals with gaps bounded by b.
Edit? What the ****!
[edit] 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 S is piecewise syndetic and
, then for some 
, Ci 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]
