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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 denoted as FS(D). Slightly more generally, for a sequence of natural numbers (ni), one can consider the set of finite sums FS((ni)), consisting of the sums of all finite length subsequences of (ni).
A set A of natural numbers is an IP set if there exists an infinite set D such that FS(D) is a subset of A. Equivalently, one may require that A contains all finite sums FS((ni)) of a sequence (ni).
Some authors give a slightly different definition of IP sets: They require that FS(D) equal A instead of just being a subset.
The term IP set was coined by Hillel Furstenberg and Benjamin Weiss[1][2] to abbreviate "infinite-dimensional parallelepiped". Serendipitously, the abbreviation IP can also be expanded to "idempotent"[3] (a set is an IP if and only if it is a member of an idempotent ultrafilter).
^Furstenberg, H.; Weiss, B. (1978). "Topological Dynamics and Combinatorial Number Theory". Journal d'Analyse Mathématique. 34: 61–85. doi:10.1007/BF02790008.
^Harry, Furstenberg (July 2014). Recurrence in ergodic theory and combinatorial number theory. Princeton, New Jersey. ISBN 9780691615363. OCLC 889248822.{{cite book}}: CS1 maint: location missing publisher (link)
^Bergelson, V.; Leibman, A. (2016). "Sets of large values of correlation functions for polynomial cubic configurations". Ergodic Theory and Dynamical Systems. 38 (2): 499–522. doi:10.1017/etds.2016.49. ISSN 0143-3857. S2CID 31083478.
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