A hypercontractive proof of the sharp bound for cancellative pairs

Fan Chang

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arXiv · 2609.09105

A hypercontractive proof of the sharp bound for cancellative pairs

Fan Chang

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Abstract

Fang and Huang proved that every cancellative pair $(\mathcal{A},\mathcal{B})$ of families of subsets of $[n]$ satisfies $|\mathcal{A}|\cdot|\mathcal{B}|\leq(\frac{9}{4})^n$. Their proof uses entropy. We give a Fourier-analytic proof based on Lifshitz's one-sided noise operator $\mathrm{T}^{1/4\to1/2}$. The main tool is a near-$L^1$ hypercontractive estimate for this operator.

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Fan Chang. 2026-09-08. A hypercontractive proof of the sharp bound for cancellative pairs. https://arxiv.org/abs/2609.09105

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