There is a well-known theorem of Freiman saying that if $A$ is a finite set of integers with $|2A| \le 3|A|-4$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms. Is there an analogue of this result for the "restricted sumset" $2^{\hat{\,}} A = \{ a+b:\,a,b\in A,\,a\ne b \} $? For the higher restricted sumsets $k^{\hat{}}A$ with $k\ge 3$?

There is a well-known theorem of Freiman saying that if $A$ is a finite set of integers with $|2A| \le 3|A|-4$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms. Is there an analogue of this result for the "restricted sumset" $2^{\hat{\,}} A = \{ a+b:\,a,b\in A,\,a\ne b \} $? How "long" can $A$ be given that $|2^{\hat{\,}} A|$ is small?