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arXiv · 2209.04243

An analogue of Bonami's Lemma for functions on spaces of linear maps, and 2-2 Games

David Ellis·Guy Kindler·Noam Lifshitz
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Abstract

We prove an analogue of Bonami's (hypercontractive) lemma for complex-valued functions on $\mathcal{L}(V,W)$, where $V$ and $W$ are vector spaces over a finite field. This inequality is useful for functions on $\mathcal{L}(V,W)$ whose `generalised influences' are small, in an appropriate sense. It leads to a significant shortening of the proof of a recent seminal result by Khot, Minzer and Safra that pseudorandom sets in Grassmann graphs have near-perfect expansion, which (in combination with the work of Dinur, Khot, Kindler, Minzer and Safra) implies the 2-2 Games conjecture (the variant, that is, with imperfect completeness).

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David Ellis, Guy Kindler, Noam Lifshitz. 2022-09-09. An analogue of Bonami's Lemma for functions on spaces of linear maps, and 2-2 Games. https://arxiv.org/abs/2209.04243

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