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Write $\delta =d_R(X,Y)$, $A=U_1+U_2$, $B=V_1+V_2$, and $S=A+B$. Distances depend only on distributions, so all variables used in any one application may be realized as independent random variables.

First, the entropy submodularity for three independent sums impliesand analogously for $V$. The Entropic Ruzsa triangle inequality gives $d_R(U,U)\le 2d_R(U,X)$ and $d_R(V,V)\le 2d_R(V,Y)$. Consequently the relevance of independent self-sums gives

Apply the Conditioned entropic Ruzsa distance of a summand first to $(A,B,X)$ and then to $(B,A,Y)$:Three applications of the Entropic Ruzsa triangle inequality giveAdding all these bounds yieldsThus the required absolute constants may be taken as $a=4$ and $b=2$.

Solved by gpt-5.6-sol high.

Because the Entropic Ruzsa distance depends only on marginal distributions, take $X,Y,Z$ independent with the required marginals. Since $X-Z=(X-Y)+(Y-Z)$ is a function of $(X-Y,Y-Z)$, the data processing inequality for mutual information yieldsThe map $(X,X-Y,Y-Z)\mapsto (X,Y,Z)$ is a bijection. Using independence and the chain rule for information entropy, the left side iswhereas the right side is $H(X-Z)-H(Z)$. Hencewhere the final step is subadditivity of information entropy. Substituting this inequality into the definition of $d_R$ gives the Entropic Ruzsa triangle inequality

Solved by gpt-5.6-sol high.

Let $X_1,X_2,Y$ be independent, with $X_1,X_2$ distributed as $X$. Apply part (b) to $X_1,-Y,X_2$:Adding an independent random variable cannot decrease information entropy, soIn terms of Entropic Ruzsa distance, this is $d_R(X,-X)\le 2d_R(X,Y)$. The Entropic Ruzsa triangle inequality and invariance under simultaneous negation now givewhich is the Entropic Ruzsa sum-difference inequality.

Solved by gpt-5.6-sol high.

If $(U,V)$ is $C$-relevant to $(X,Y)$ and $U_1,U_2,V_1,V_2$ are mutually independent copies, then $(U_1+U_2,V_1+V_2)$ is $2C$-relevant to $(X,Y)$. The entropy submodularity for three independent sums impliesand similarly for $V$. The Entropic Ruzsa triangle inequality boundswhich proves the claim after addition.