Abstract:
We consider the following inequality relating the entropy h(p) of a Bernoulli r.v. of a single positive outcome with the entropy h(p^k) of the corresponding r.v. of k positive outcomes:
\phi^{k-1}h(p^k) \ge p^{k-1}h(p) where \phi is the positive root of x^k+x-1.
The case k=2 of this inequality was recently proved by Alweiss, Huang, and Sellke and by Boppana, and served, together with a breakthrough idea of Gilmer, an important ingredient in the proof of the stability version of Frankl's union-closed conjecture by Chase and Lovett. Other applications of this inequality were recently considered by Wakhare.
We prove this inequality for k=3,4, and prove a variant for all k where \phi^{k-1} is replaced with a slightly larger constant. In fact, we show that the inequality reduces to a conjecture about the number of real roots of an explicit polynomial. We use this result to prove the k-dimensional analogue of the Chase-Lovett theorem.
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