On linear-combinatorial problems associated with subspaces spanned by $\{\pm 1\}$-vectors

TL;DR

This work determines the probability that a random subspace spanned by $p$ {±1}-vectors in $\mathbb{R}^n$ contains another {±1}-vector, showing the event is dominated by triple interactions. The authors develop an $\eta^{\bigstar}_n$ framework that links this probability to $P(n,n+1)$ and to the fraction of singular Bernoulli matrices, and prove the main asymptotic $P(p,n)=P_3(p,n)+O((5/8+o_n(1))^n)$. The proof uses a decomposition into $P_m(p,n)$, a three-case analysis (3.1–3.3), and LL0-type bounds together with rank/space counting to bound subleading contributions. The results connect random subspace geometry with threshold-function counts and the asymptotics of singular $\{\\pm1\\}$-matrices, extending prior work of Odlyzko, Kalai–Linial, and related random-matrix analyses.

Abstract

A complete answer to the question about subspaces generated by $\{\pm 1\}$-vectors, which arose in the work of I.Kanter and H.Sompolinsky on associative memories, is given. More precisely, let vectors $v_1, \ldots , v_p,$ $p\leq n-1,$ be chosen at random uniformly and independently from $\{\pm 1\}^n \subset {\bf R}^n.$ Then the probability ${\mathbb P}(p, n)$ that $$span \ \langle v_1, \ldots , v_p \rangle \cap \left\{ \{\pm 1\}^n \setminus \{\pm v_1, \ldots , \pm v_p\}\right\} \ne \emptyset \ $$ is shown to be $$4{p \choose 3}\left(\frac{3}{4}\right)^n + O\left(\left(\frac{5}{8} + o_n(1)\right)^n\right) \quad \mbox{as} \quad n\to \infty,$$ where the constant implied by the $O$-notation does not depend on $p$. The main term in this estimate is the probability that some 3 vectors $v_{j_1}, v_{j_2}, v_{j_3}$ of $v_j$, $j= 1, \ldots , p,$ have a linear combination that is a $\{\pm 1\}$-vector different from $\pm v_{j_1}, \pm v_{j_2}, \pm v_{j_3}. $

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Theorem 2.2
• Corollary 2.3
• Theorem 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3