#### Alladi Ramakrishnan Hall

#### Improved Bounds on Fourier entropy and Min-entropy

#### Nitin Saurabh

##### MPII --> Technion

*Given a Boolean function f : {-1,1}^n -> {-1,1}, define the Fourier distribution to be the distribution on subsets of [n], where each subset S of [n] is sampled with probability equaling square of the Fourier coefficient associated with S, \hat{f}(S)^2. The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai (1996) seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C>0 such that H(f) <= C Inf(f), where H(f) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f.*

We will discuss the significance of the conjecture, present new upper bounds on Fourier entropy of Boolean functions and also discuss its relationship with Mansour's conjecture.

This talk is based on a joint work with S. Arunachalam, S. Chakraborty, M. Koucky, and R. de Wolf.

Done