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We show that there is an absolute 
$c>0$ such that if 
$A$ is a finite set of integers, then there is a set 
$S\subset A$ of size at least 
$\log ^{1+c}|A|$ such that the restricted sumset 
$\{s+s^{\prime }:s,s^{\prime }\in S\text{ and }s\neq s^{\prime }\}$ is disjoint from 
$A$. (The logarithm here is to base 
$3$.)
We prove quantitative versions of the Balog–Szemerédi–Gowers and Freiman theorems in the model case of a finite field geometry 𝔽2n, improving the previously known bounds in such theorems. For instance, if 
is such that ∣A+A∣≤K∣A∣ (thus A has small additive doubling), we show that there exists an affine subspace H of 𝔽2n of cardinality 
such that 
. Under the assumption that A contains at least ∣A∣3/K quadruples with a1+a2+a3+a4=0, we obtain a similar result, albeit with the slightly weaker condition ∣H∣≫K−O(K)∣A∣.
