Talk information

Date: Sunday, November 3, 2024
Time: 10:10–11:00
Place: Schreiber 309
Speaker: Raphael Yuster (University of Haifa)
Title: An entropy inequality and almost $k$-union closed set systems

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 all $3\le k\le 20$ 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.

Part of this work is joint with Geva Yashfe.