Bad Science Matrices

Inspired by the bad scientist who keeps repeating an experiment 20 times to get a single outcome with $p < 0.05$, we consider matrices $A \in \mathbb{R}^{n \times n}$ whose rows are normalized in $\ell^2$ and for which $2^{-n}\sum_{x \in \left\{-1,1\right\}^n} \|Ax\|_{\ell^{\infty}}$ is large. They correspond to affine transformations of the discrete unit cube to points with, on average, at least one large coordinate. Such matrices can be seen as a collection of fair tests on a fair coin where at least one outcome is typically atypical. We prove that, as $n \rightarrow \infty$, the quantity can scale as $$\max_{A \in \mathbb{R}^{n \times n}} \frac{1}{2^{n}}\sum_{x \in \left\{-1,1\right\}^n} \|Ax\|_{\ell^{\infty}} = (1+o(1)) \cdot \sqrt{2\log{n}}.$$ We also present candidate maximizers up to dimension $n \leq 8$ which appear to be highly structured and have nice closed-form solutions.