Linear Independence of Random Boolean Tensor Powers at the Dimension Threshold

Let $d\ge 1$ be fixed and let $$D(n,d):=\sum _{j=0}^d\left(\genfrac{}{}{0px}{}{n-1}{j}\right)\mathrm{.}$$ We show that if $x^{(1)},\dots ,x^{(m)}$ are independent uniform points of $\{\pm 1{\}}^n$ then uniformly for $m\le D(n,d)$, there exists a constant $C_d>0$ such that $$\mathbb{P}((x^{(1)}{)}^{\otimes d},\dots ,(x^{(m)}{)}^{\otimes d}\text{ are linearly independent})=1-O_d\left(\frac{{\log }^{C_d}n}{n^{1\mathrm{/}2}}\right)\mathrm{.}$$ This achieves the exact dimensional threshold and answers a question asked by Baldi and Vershynin.