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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.

Write $S_1=H(X)+H(Y)+H(Z)$ and $S_2=H(X,Y)+H(Y,Z)+H(Z,X)$. By the definition of mutual information,so the required right-hand side is $(2S_2-S_1)/3$.

Apply entropy submodularity to the pairs $(X,Y),(X,Z)$ and then cyclically permute the variables:Adding gives $2S_2\ge S_1+3H(X,Y,Z)$, which is exactly

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The axioms for information entropy give the formula $H(X)=-{\sum }_{x}p(x)\log p(x)$ and hence the chain rule for information entropyBecause conditioning reduces entropy, $H(Y\mid X)\le H(Y)$, and thereforeThis is subadditivity of information entropy.

For the entropy submodularity rule, apply the chain rule for information entropy twice:The last quantity is conditional mutual information, whose nonnegativity again expresses that conditioning reduces entropy.

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The Entropic Balog-Szemerédi-Gowers theorem states that for finitely supported random variables $A,B$ in an abelian group,where the left-hand side is the Simultaneous conditional entropic Ruzsa distance.

Put $S=A+B$ and take two copies $(A_1,B_1)$ and $(A_2,B_2)$ that are conditionally independent given $S$. Thus both sums equal $S$, and, given $S$, $A_1$ and $B_2$ are independent with the required conditional marginals. By conditioning reduces entropy,The two conditional entropies are equal, since either $A$ or $B$ together with $S$ determines the other, and the chain rule for information entropy gives

Set $D=A_1-B_2$. By entropy submodularity,The first term is $H(A_1,B_2)\le H(A)+H(B)$ by subadditivity of information entropy. Since $A_1+B_1=A_2+B_2$, we also have $D=A_2-B_1$, so the second term is at most $H(A)+H(B)$. Finally $(D,A_1,B_1)$ determines all four copied variables, and conditional independence givesAs $H(A,B)=H(A)+H(B)-I(A;B)$, these estimates implySubtracting the common conditional-entropy term proves the stated Entropic Balog-Szemerédi-Gowers theorem.

Solved by gpt-5.6-sol high.