In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers. A set S ⊂ N {\displaystyle S\subset \mathbb {N} } is called piecewise syndetic if there exists a finite subset G of N {\displaystyle \mathbb {N} } such that for every finite subset F of N {\displaystyle \mathbb {N} } there exists an x ∈ N {\displaystyle x\in \mathbb {N} } such that
x + F ⊂ ⋃ n ∈ G ( S − n ) {\displaystyle x+F\subset \bigcup _{n\in G}(S-n)}
where S − n = { m ∈ N : m + n ∈ S } {\displaystyle S-n=\{m\in \mathbb {N} :m+n\in S\}} . Equivalently, S is piecewise syndetic if there is a constant b such that there are arbitrarily long intervals of N {\displaystyle \mathbb {N} } 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 β N {\displaystyle \beta \mathbb {N} } , the Stone–Čech compactification of the natural numbers. Partition regularity: if S {\displaystyle S} is piecewise syndetic and S = C 1 ∪ C 2 ∪ ⋯ ∪ C n {\displaystyle S=C_{1}\cup C_{2}\cup \dots \cup C_{n}} , then for some i ≤ n {\displaystyle i\leq n} , C i {\displaystyle C_{i}} contains a piecewise syndetic set. (Brown, 1968) If A and B are subsets of N {\displaystyle \mathbb {N} } with positive upper Banach density, then A + B = { a + b : a ∈ A , b ∈ B } {\displaystyle A+B=\{a+b:a\in A,\,b\in B\}} is piecewise syndetic.
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
Notes
References McLeod, Jillian (2000). "Some Notions of Size in Partial Semigroups" (PDF). Topology Proceedings. 25 (Summer 2000): 317–332. Bergelson, Vitaly (2003). "Minimal Idempotents and Ergodic Ramsey Theory" (PDF). Topics in Dynamics and Ergodic Theory. London Mathematical Society Lecture Note Series. Vol. 310. Cambridge University Press, Cambridge. pp. 8–39. doi:10.1017/CBO9780511546716.004. ISBN 978-0-521-53365-2. Bergelson, Vitaly; Hindman, Neil (2001). "Partition regular structures contained in large sets are abundant". Journal of Combinatorial Theory. Series A. 93 (1): 18–36. doi:10.1006/jcta.2000.3061. Brown, Thomas Craig (1971). "An interesting combinatorial method in the theory of locally finite semigroups". Pacific Journal of Mathematics. 36 (2): 285–289. doi:10.2140/pjm.1971.36.285.
