Lucky Talk: An Information Theoretic View of Hypercontractivity
summary
Proves a fractional Shearer inequality for the t=2 case by partitioning the ground set, encoding weighted Fourier coefficients as integer multiplicities, and equating logs with entropies of a deliberately constructed random variable that carries x plus four index counters; uses the chain rule to cancel one copy of each coordinate entropy and the cover-two hypothesis to cancel the other, yielding the desired bound. Also gives an information theoretic proof of Bonami Beckner via a three variable x y z support built from blocks of sizes r and s with r/(r+s) = (1+epsilon)/2 and reduces the remaining step to a symmetric calculus minimization solved by Lagrange multipliers.
AI-written summary · about