Let $A$ be a finite set of integers with $|A \hat{+} A| \leq K|A|$, where the $\hat{+}$ denotes restricted sumset: the set of all $a\_1 + a\_2$ with $a\_1, a\_2 \in A$ and $a\_1 \neq a\_2$. 


Claim: $|A + A| \leq (K + o(1))|A|$, where $o(1)$ denotes a quantity tending to 0 as $|A|$ tends to $\infty$.


Sketch proof: Let $A'$ be the set of all $a \in A$ for which $2a$ cannot be represented as $a\_1 + a\_2$ with $a\_1, a\_2 \in A$, $a\_1 \neq a\_2$. Note that $2A' = (A + A) \setminus (A \hat{+} A)$, so it suffices to show that $A'$ is small. But $A'$ has no 3-term arithmetic progressions. But by standard results (Freiman, Ruzsa...) a set $A'$ with no 3-term progressions has $|A' + A'| \gg |A'| \log^c |A'|$. Thus indeed $|A'| = o(|A|)$. 


I know I've seen this argument in the literature, and my question is simply: where?



Additive Combinatorics - reference request
For a somewhat similar argument, see Proposition 2.5 from Alon's 1987 paper ``Subset sums", available [here](http://www.tau.ac.il/~nogaa/PDFS/Publications/Subset%20sums.pdf). 


