Towards a Proof of the Fourier-Entropy Conjecture?

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is $$o(\log n)$$ . However, both results become useless when the total influence of the function is $$\omega (\log n)$$ . The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of $$S_n$$ . In this paper, we build and improve on the techniques of the Bourgain–Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include: Our concentration result for the Fourier spectrum of functions with small total influence also has new implications in learning theory. More specifically, we conclude that the class of functions whose total influence is at most K is agnostically learnable in time $$2^{O(K\log K)}$$ using membership queries. Thus, the class of functions with total influence $$O(\log n/\log \log n)$$ is agnostically learnable in $$\mathsf{poly}(n)$$ time.