Three Standard Deviations Suffice While One Does Not

Abstract

Spencer's 1985 ``six standard deviations suffice'' theorem shows that every $A \in [-1,1]^{n \times n}$ has a sign vector $x \in \{-1,1\}^n$ with $\|Ax\|_\infty \le 6\sqrt{n}$. We show the upper bound $\sqrt{3\operatorname{arsinh}(10)}\sqrt{n}+4 < 2.9992 \sqrt{n} + 4$ by directly rounding the minimizer of a potential function to a vertex of the cube. We also show that, for every power of two $n \ge 2^{50}$, there exists a matrix $A \in \{-1,1\}^{n \times n}$ such that $\|Ax\|_\infty >1.0000002\sqrt{n}$ for every choice of signs $x \in \{-1,1\}^n$. The construction simply replaces a $2^{-22}$ fraction of the columns of a Hadamard matrix with independent random sign vectors. This is the first improvement over the $\sqrt{n}$ lower bound of Olson and Spencer (1978), which uses a Hadamard matrix.