A hypercontractive proof of the sharp bound for cancellative pairs

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