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

For random variables $A,B$ in a finite additive group, take independent copies $A^{\prime },B^{\prime }$ with the same respective distributions and define the Entropic Ruzsa distance byFor the independent variables in the question, expansion givesPart iii, applied to the independent variables $X,Y,-Z$, saysSubtracting $H(X-Y)$ from both sides proves

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