For random variables $A,B$ in a <finite additive group>, take independent copies $A',B'$ with the same respective distributions and define the <Entropic Ruzsa distance> by
$$
d_R(A,B)=H(A'-B')-\frac12H(A')-\frac12H(B').
$$
For the independent variables in the question, expansion gives
$$
d_R(X,Z)+d_R(Y,Z)-d_R(X,Y)
=H(X-Z)+H(Y-Z)-H(X-Y)-H(Z).
$$
Part iii, applied to the independent variables $X,Y,-Z$, says
$$
H(X+Y-Z)+H(Z)\leq H(X-Z)+H(Y-Z).
$$
Subtracting $H(X-Y)$ from both sides proves
$$
H(X+Y-Z)-H(X-Y)
\leq d_R(X,Z)+d_R(Y,Z)-d_R(X,Y).
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 164 / 1 / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-164/1/iv/solution