For every discrete real-valued random variable X with finite entropy, max{H(X+X'), H(XX')} ≥ (8/7)H(X) - O(log H(X)).
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The Entropic Sum-Product Phenomenon
For every discrete real-valued random variable X with finite entropy, max{H(X+X'), H(XX')} ≥ (8/7)H(X) - O(log H(X)).