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Let $A_1$ and $A_2$ be randomly chosen subsets of the first $n$ integers ofcardinalities $s_2\geq s_1 = \Omega(s_2)$, such that their sumset $A_1+A_2$ hassize $m$. We show that asymptotically almost surely $A_1$ and $A_2$ are almostfully contained in arithmetic progressions $P_1$ and $P_2$ with the same commondifference and cardinalities approximately $s_i m/(s_1+s_2)$. We also prove acounting theorem for such pairs of sets in arbitrary abelian groups. Theresults hold for $s_i = \omega(\log^3 n)$ and $s_1+s_2 \leq m = o(s_2/\log^3n)$. Our main tool is an asymmetric version of the method of hypergraphcontainers which was recently used by Campos to prove similar results in thespecial case $A=B$.

The typical approximate structure of sets with bounded sumset