Affine Subspace Statistics in the Hypercube

TL;DR

This paper investigates the intersection behavior of affine subspaces in high-dimensional hypercubes, revealing how the maximum fraction of intersections depends on the size and structure of the subset.

Contribution

It introduces new asymptotic results and bounds for the intersection statistics of affine subspaces in hypercubes, extending prior work on axis-aligned subcubes.

Findings

For s = j·2^k with j odd, λ*(d,s) approaches 1 - Θ(2^{-k}) as d increases.

When s is odd, λ*(d,s) is at most 1/2, differing from axis-aligned subcube behavior.

Provides bounds for specific intersection sizes s.

Abstract

We study the intersection statistics of affine subspaces in the hypercube $\mathbb{F}_2^n$, motivated by recent work of Alon, Axenovich, and Goldwasser on the intersection statistics of axis-aligned subcubes of an $n$-dimensional cube. Let $d\ge 1$ and $0\le s\le 2^d$ be nonnegative integers. For a subset $A\subseteq \mathbb{F}_2^n$ where $n\ge d$, define $\lambda^*(n,d,s,A)$ to be the fraction of affine $d$-flats in $\mathbb{F}_2^n$ that intersect $A$ at exactly $s$ points. Let $\lambda^*(n,d,s) = \max_{A\subseteq \mathbb{F}_2^n}\lambda^*(n,d,s,A)$ and let $\lambda^*(d,s) = \lim_{n\to \infty}\lambda^*(n,d,s)$. We show that when $s = j\cdot 2^k$ with $j$ odd and $k\ge 1$, we have $\lambda^*(d,s)\to 1-\Theta(2^{-k})$ as $d\to \infty$. This implies that $\lambda^*(d,s)$ is controlled up to constant factors by the $2$-adic valuation of $s$ when $s$ is even. When $s$ is odd, we show that $\lambda^*(d,s)\le \frac{1}{2}$ in contrast to the behavior of axis-aligned subcube statistics. We also present several upper and lower bounds for certain specific values of $s$.