It is a basic folklore fact from the area of additive combinatorics that a subset $A$ of an abelian group satisfies $|2A|<\frac32\,|A|$ if and only if $A$ is contained in a coset of a (finite) subgroup $H$ with the density $|A|/|H|>\frac23$. What are the historical origins of this fact? It was certainly known to Freiman around years 1961-1965, but I suspect that it can appear somewhere in the earlier papers by Kemperman / Scherk / Mann / Olson.
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1$\begingroup$ terrytao.files.wordpress.com/2013/11/expander-book.pdf In this book, Theorem 1.5.2, Tao attributes it to a 73 paper of Freiman. $\endgroup$– George ShakanJul 21, 2020 at 9:16
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$\begingroup$ @GeorgeShakan: It is true that Theorem 1.5.2 from Tao's book is presented in Freiman's [1973b], but I am by far not sure that the abelian version of this theorem had not appeared elsewhere before. $\endgroup$– SevaJul 21, 2020 at 9:39
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