Polynomial representation, Fourier transform and complexity of Boolean functions

We study the Log-rank Conjecture for XOR functions and the Sensitivity Conjecture by exploiting structure of the Fourier transform and polynomial representations of Boolean functions. For the Log-rank Conjecture for XOR functions, we show that the it is true for functions with small spectral norm or low F2-degree. More precisely, we prove the following two bounds. 1. CC(f(x⊕ y)) = O(‖f̂‖1 · log ‖f̂‖0), and 2. CC(f(x⊕ y)) = O(2d2/2 logd−1 ‖f̂‖0), where d = degF2(f). For the Sensitivity Conjecture, we obtain a pair of conjectures with the property that (i). each of them is a consequence of the Sensitivity Conjecture, (ii). neither one of them is known to imply the Sensitivity Conjecture and (iii) both of them together imply the Sensitivity Conjecture. We also obtain a new sufficient condition for the Sensitivity Conjecture based on a graph that we called the monomial graph associated with the Boolean function. Specifically, we show that in order to prove the Sensitivity Conjecture, it suffices to show that for all Boolean functions f , the number of monotone paths from 0 to some level ` = Ω(n) in its monomial graph is at most s(f). On the other hand, we show that for all Boolean functions f , 1. the number of monotone paths from 0 to level ` in its monomial graph is at most min{2`2/2s(f)`, (s(f)4)}; 2. the number of degree-` monomials in its R-representation is at most (4e)`s(f)`·min{s(f),`}.