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

TL;DR

This paper extends Bonami's hypercontractive lemma to functions on spaces of linear maps over finite fields, enabling simplified proofs of key results in pseudorandomness and the 2-2 Games conjecture.

Contribution

It introduces a new hypercontractive inequality for functions on linear maps, facilitating shorter proofs of important results in combinatorics and computational complexity.

Findings

Proves an analogue of Bonami's lemma for functions on $\\mathcal{L}(V,W)$.

Simplifies the proof of near-perfect expansion of pseudorandom sets in Grassmann graphs.

Contributes to the proof of the 2-2 Games conjecture with imperfect completeness.

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).