= Relevance of independent self-sums
Let $U,V,X,Y$ take values in the <vector space> $\mathbb F_2^n$ over the two-element <finite field>. In this <additive group>, subtraction equals addition. Suppose $(U,V)$ is a <C-relevant pair of random variables> relative to $(X,Y)$, and take mutually <independent> copies $U_1,U_2,V_1,V_2$. Then $(U_1+U_2,V_1+V_2)$ is $2C$-relevant to $(X,Y)$.
For a proof, let $S=U_1+U_2$ and take an <independent> representative of $X$. The <entropy submodularity for three independent sums> gives
$$
H(S+X)+H(U)\leq H(S)+H(U+X),
$$
so the definition of <Entropic Ruzsa distance> yields
$$
d_R(S;X)\leq d_R(U;X)+\tfrac12d_R(U;U)\leq2d_R(U;X).
$$
The last step is the <Entropic Ruzsa triangle inequality>. Apply the same argument to $V_1+V_2,Y$ and add the inequalities.<On a conjecture of Marton by Timothy Gowers and others>[Appendix A, equation (A.13) and Lemma A.1, printed pp. 27–28]{ref}
The binary <finite field> hypothesis makes the displayed self-sum argument agree with the difference convention in <Entropic Ruzsa distance>. On a general <abelian group>, signs must be retained explicitly.
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