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 ...

Fix coprime s, t > 1. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the finitely many simultaneous (s, t)cores have average size 1 24(s−1)(t−1)(s+t+1), and that the subset of self-conjugate cores has the same average (first shown by Chen–Huang–Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse...