Piecewise syndetic set

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In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers.

A set S \sub \mathbb{N} is called piecewise syndetic if there exists a finite subset G of \mathbb{N} such that for every finite subset F of \mathbb{N} there exists an x \in \mathbb{N} such that

x+F \subset \bigcup_{n \in G} (S-n)

where S-n = \{m \in \mathbb{N}: m+n \in S \}. Equivalently, S is piecewise syndetic if there are arbitrarily long intervals of \mathbb{N} where the gaps in S are bounded by some constant b.

Properties[edit]

  • 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 \beta \mathbb{N}, the Stone–Čech compactification of the natural numbers.
  • Partition regularity: if S is piecewise syndetic and S = C_1 \cup C_2 \cup ... \cup C_n, then for some i \leq n, C_i contains a piecewise syndetic set. (Brown, 1968)

Other Notions of Largeness[edit]

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

See also[edit]

Notes[edit]

  1. ^ R. Jin, Nonstandard Methods For Upper Banach Density Problems, Journal of Number Theory 91, (2001), 20-38</math>.

References[edit]