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

David Ellis, Guy Kindler, Noam Lifshitz

TL;DR

This work develops a Bonami-type (hypercontractive) inequality for complex-valued functions on the bilinear space $\mathcal{L}(V,W)$ over finite fields, with a focus on global functions (small generalized influences). The authors establish a conditional hypercontractive inequality that bounds the $L^4$ norm of degree-$d$ functions by the $L^2$ norms of their derivatives, enabling a streamlined proof of small-set expansion for pseudorandom sets in the Grassmann graph and, consequently, progress toward the 2-2 Games conjecture. A key innovation is the junta/Fourier-degree framework for $\mathcal{L}(V,W)$, the introduction of $d$-restrictions, and a rich calculus of (hybrid) Laplacians and derivatives that manage degree reduction without dimension blow-up. These results connect hypercontractivity, restriction-globalness, and generalized influences to yield near-optimal small-set expansion theorems, providing a simpler pathway to the Grassmann Expansion Hypothesis and its cryptographic and complexity-theoretic implications. The framework also offers a blueprint for extending similar techniques to other groups and schemes, including applications to finite simple groups via bilinear embeddings and to related expansion problems in coding theory and probabilistic method settings.

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).

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

TL;DR

This work develops a Bonami-type (hypercontractive) inequality for complex-valued functions on the bilinear space over finite fields, with a focus on global functions (small generalized influences). The authors establish a conditional hypercontractive inequality that bounds the norm of degree- functions by the norms of their derivatives, enabling a streamlined proof of small-set expansion for pseudorandom sets in the Grassmann graph and, consequently, progress toward the 2-2 Games conjecture. A key innovation is the junta/Fourier-degree framework for , the introduction of -restrictions, and a rich calculus of (hybrid) Laplacians and derivatives that manage degree reduction without dimension blow-up. These results connect hypercontractivity, restriction-globalness, and generalized influences to yield near-optimal small-set expansion theorems, providing a simpler pathway to the Grassmann Expansion Hypothesis and its cryptographic and complexity-theoretic implications. The framework also offers a blueprint for extending similar techniques to other groups and schemes, including applications to finite simple groups via bilinear embeddings and to related expansion problems in coding theory and probabilistic method settings.

Abstract

We prove an analogue of Bonami's (hypercontractive) lemma for complex-valued functions on , where and are vector spaces over a finite field. This inequality is useful for functions on 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).
Paper Structure (42 sections, 64 theorems, 124 equations)

This paper contains 42 sections, 64 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Warm up: Hypercontractivity on $\mathbb{F}_{p}^{n}$
  3. The Fourier expansion of functions on $\mathbb{F}_{p}^{n}$
  4. Restrictions
  5. Global functions
  6. The Efron-Stein decomposition
  7. Laplacians and their Efron-Stein formulae
  8. Derivatives
  9. Influences
  10. Overview of the proof of Theorem \ref{['thm:hyp-prod']}
  11. Proof of Theorem \ref{['thm:hyp-prod']}
  12. Our hypercontractive inequality for global functions
  13. Overview of the proof of small-set expansion in $\mathbb{F}_p^n$.
  14. Obtaining small-set expansion
  15. Fourier analysis on $\mathcal{L}(V,W)$
  16. ...and 27 more sections

Key Result

Lemma 1.1

Let $f:\{0,1\}^n \to \mathbb{R}$ be a function of degree at most $d$; then

Theorems & Definitions (73)

  • Lemma 1.1: Bonami's Lemma
  • Theorem 1.4: Keevash, Lifshitz, Long, Minzer, 2019+
  • Theorem 1.5
  • Theorem 1.6: Small-set expansion theorem for the shortcode graph
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Proposition 2.6
  • ...and 63 more