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

Table of contents
- Solution iv

Solution

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^{'}) - \frac{1}{2} H (A^{'}) - \frac{1}{2} H (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.

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