Stability of near-extremizers for the high-entropy increment

Characterize the additive structure forced when the high-entropy increment H(X+Y) - (H(X)+H(Y))/2 approaches 1/2, including the effects of additive embeddings into the integers and near-equality cases of the continuous entropy power inequality.

Background

The paper proves that the asymptotically sharp additive constant in the entropy lower bound is 1/2 for independent discrete random variables in torsion-free abelian groups and, under an entropy-deficit condition, in prime cyclic groups. Binomial distributions attain this constant asymptotically, but the authors note that this example may not describe all near-extremizers.

The proof reduces finite-support distributions to integer-valued variables through additive embeddings that preserve only finitely many additive relations. Consequently, a stability theorem would need to identify which additive structures are necessarily present when the entropy increment is close to the optimal value, while also incorporating near-equality structure from the continuous entropy power inequality.