Information Theory and Additive Combinatorics
summary
Shows two concrete claims: on Z and R there exist IID finite-entropy variables for which H(X−X′) − H(X+X′) is arbitrarily large gap, yet after subtracting H(X) the entropy increases for sum vs difference have a universal bound of ratio between 1/2 and 2 for any locally compact abelian group (uses Haar measure and Rényi/differential/Shannon entropies). Proofs use combinatorial sumset ideas, approximation from integers to reals, and a discrete integer rearrangement inequality that recovers Cauchy–Davenport.
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