An FKN Theorem for the Binary Grassmann Scheme

2026-08-11

AI Summary

This work extends the classical Friedgut–Kalai–Naor (FKN) theorem to the binary Grassmann scheme, investigating the structural properties of Boolean functions defined on the set of ℓ-dimensional subspaces that are close to degree-one functions. By combining combinatorial arguments, Fourier analysis, and finite field geometry, the authors establish the first FKN-type structure theorem in this setting, showing that any such Boolean function—or its complement—must be close to a simple function determined by a finite collection of point and hyperplane indicators. This result reveals an intrinsic rigidity of approximately linear Boolean functions over high-dimensional subspaces and significantly broadens the understanding of structural properties of Boolean functions beyond classical domains.

Abstract

A classical theorem due to Friedgut, Kalai and Naor asserts that if a function $f\colon \{0,1\}^n\to\{-1,1\}$ close to a degree $1$ function, then either $f$ or $-f$ is close to either the all $1$ function, or to $(-1)^{x_i}$ for some $i\in [n]$. We prove a version of their theorem for the Grassmann scheme over $\mathbb{F}_2$. More precisely, we prove if a function $f\colon \genfrac{[}{]}{0pt}{}{\mathbb{F}_2^n}{\ell}\to\{0,1\}$ is close to a degree $1$ function, then either $f$ or $1-f$ must be close to a function of the form $g(L) = \sum_{x\in\mathcal{X}}1_{x\in L}+\sum_{W\in\mathcal{W}}1_{L\subseteq W}$, where $\mathcal{X}\subseteq\mathbb{F}_2^n$ is a set of points and $\mathcal{W}$ is a set of hyperplanes in $\mathbb{F}_2^n$.

Problem

• FKN Theorem
• Grassmann Scheme
• Boolean Functions
• Degree 1 Functions
• Fourier Analysis

Innovation

• FKN theorem
• Grassmann scheme
• Boolean functions
• degree-1 approximation
• hyperplane indicators