Let $X_1,X_2,Y$ be independent, with $X_1,X_2$ distributed as $X$. Apply part (b) to $X_1,-Y,X_2$:
$$
H(X_1+X_2-Y)+H(Y)\leq2H(X-Y).
$$
Adding an <independent random variable> cannot decrease <information entropy>, so
$$
H(X_1+X_2)+H(Y)\leq2H(X-Y).
$$
In terms of <Entropic Ruzsa distance>, this is $d_R(X,-X)\leq2d_R(X,Y)$. The <Entropic Ruzsa triangle inequality> and invariance under simultaneous negation now give
$$
d_R(X,-Y)
\leq d_R(X,-X)+d_R(-X,-Y)
\leq2d_R(X,Y)+d_R(X,Y),
$$
which is the <Entropic Ruzsa sum-difference inequality>.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-224/1/d/solution