In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, piecewise syndeticity is a notion of largeness of subsets of the natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

s.

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 S contains arbitrarily long intervals with gaps bounded by some fixed bound b.

Properties

• A set is piecewise syndetic if and only if it is the intersection of a syndetic set
  Syndetic set
  In mathematics, a syndetic set is a subset of the natural numbers, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded.-Definition:...
  
   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
  Stone–Cech compactification
  In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX...
  
   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
  Natural density
  In number theory, asymptotic density is one of the possibilities to measure how large a subset of the set of natural numbers is....
  
  , then is piecewise syndetic

Other Notions of Largeness

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

• cofinite
  Cofinite
  In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X...
  
  
• positive upper density
• syndetic
  Syndetic set
  In mathematics, a syndetic set is a subset of the natural numbers, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded.-Definition:...
  
  
• thick
• member of a nonprincipal ultrafilter
  Ultrafilter
  In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
  
  
• IP set
  IP set
  In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set.The finite sums of a set D of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonempty subset of D.The set of all finite sums over D is often...