A hypercontractive proof of the sharp bound for cancellative pairs

Abstract

Fang and Huang proved that every cancellative pair $(\mathcal{A},\mathcal{B})$ of families of subsets of $[n]$ satisfies $|\mathcal{A}|\cdot |\mathcal{B}|\le (\frac{9}{4}{)}^n$. Their proof uses entropy. We give a Fourier-analytic proof based on Lifshitz's one-sided noise operator ${\mathrm{T}}^{1/4\to 1/2}$. The main tool is a near-$L^1$ hypercontractive estimate for this operator.

12 pages, comments welcome!