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| In [[mathematics]], '''Freiman's theorem''' is a [[combinatorial]] result in [[number theory]]. In a sense it accounts for the approximate structure of sets of [[integer]]s that contain a high proportion of their internal sums, taken two at a time.
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| The formal statement is:
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| Let ''A'' be a finite set of integers such that the [[sumset]]
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| :<math>A + A\,</math>
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| is small, in the sense that
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| :<math>|A + A| < c|A|\,</math> | |
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| for some constant <math>c</math>. There exists an [[generalized arithmetic progression|''n''-dimensional arithmetic progression]] of length
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| :<math>c' |A|\,</math> | |
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| that contains ''A'', and such that ''c''' and ''n'' depend only on ''c''.<ref>Nathanson (1996) p.251</ref>
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| A simple instructive case is the following. We always have
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| :<math>|A + A|\,</math> <span style="font-size:125%"> ≥ </span> <math>2|A|-1\,</math>
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| with equality precisely when ''A'' is an arithmetic progression.
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| This result is due to [[Gregory Freiman]] (1964,1966).<ref>Nathanson (1996) p.252</ref> Much interest in it, and applications, stemmed from a new proof by [[Imre Z. Ruzsa]] (1994).
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| ==See also==
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| *[[Markov spectrum]]
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| ==References==
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| {{reflist}}
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| * {{cite journal | zbl=0163.29501 | last=Freiman | first=G.A. | authorlink=Gregory Freiman | title=Addition of finite sets | language=English. Russian original | journal=Sov. Math., Dokl. | volume=5 | pages=1366–1370 | year=1964 }}
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| * {{cite book | first=G. A. | last=Freiman | authorlink=Gregory Freiman | title=Foundations of a Structural Theory of Set Addition | language=Russian | publisher=Kazan Gos. Ped. Inst. | location=Kazan | year=1966 | pages=140 | zbl=0203.35305 }}
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| * {{cite journal | first=G. A. | last=Freiman | authorlink=Gregory Freiman | title=Structure theory of set addition | journal=Astérisque | volume=258 | year=1999 | pages=1–33 | zbl=0958.11008 }}
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| *{{cite book| last=Nathanson | first=Melvyn B. | year=1996 | title=Additive Number Theory: Inverse Problems and Geometry of Sumsets | volume=165 | series=[[Graduate Texts in Mathematics]] | publisher=Springer | isbn=0-387-94655-1 | zbl=0859.11003 }}
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| * {{cite journal | first=Imre Z. | last=Ruzsa | authorlink=Imre Z. Ruzsa | title=Generalized arithmetical progressions and sumsets | journal=Acta Mathematica Hungarica | volume=65 | number=4 | year=1994 | pages=379–388 | zbl=0816.11008 }}
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| {{PlanetMath attribution|id=4304|title=Freiman's theorem}}
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| [[Category:Sumsets]]
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| [[Category:Theorems in number theory]]
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