Entropic Balog-Szemerédi-Gowers theorem

For finitely supported random variables $A,B$ in an abelian group,To prove it, take two conditionally independent copies $(A_1,B_1)$ and $(A_2,B_2)$ of $(A,B)$ given $A+B$. Since $A_1+B_1=A_2+B_2$, entropy submodularity givesThe first two terms are at most $H(A)+H(B)$. The last joint entropy isConsequently $H(A_1-B_2)\le H(A+B)+2I(A;B)$. Subtractingfrom this bound proves the theorem.