Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 1 / d

Table of contents
- Solution d

Solution

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}

:

H (X_{1} + X_{2} - Y) + H (Y) \leq 2 H (X - Y) .

Adding an independent random variable cannot decrease information entropy, so

H (X_{1} + X_{2}) + H (Y) \leq 2 H (X - Y) .

In terms of Entropic Ruzsa distance, this is

d_{R} (X, - X) \leq 2 d_{R} (X, Y)

. The Entropic Ruzsa triangle inequality and invariance under simultaneous negation now give

d_{R} (X, - Y) \leq d_{R} (X, - X) + d_{R} (- X, - Y) \leq 2 d_{R} (X, Y) + d_{R} (X, Y),

which is the Entropic Ruzsa sum-difference inequality.

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