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 ${\mathrm{\ell }}^2$ and for which $2^{-n}{\sum }_{x\in {\{-1,1\}}^n}\mathrm{\parallel }Ax{\mathrm{\parallel }}_{{\mathrm{\ell }}^{\mathrm{\infty }}}$ 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 $n\to \mathrm{\infty }$, the quantity can scale as $$\underset{A\in {\mathbb{R}}^{n\times n}}{\max }\frac{1}{2^n}\sum _{x\in {\{-1,1\}}^n}\mathrm{\parallel }Ax{\mathrm{\parallel }}_{{\mathrm{\ell }}^{\mathrm{\infty }}}=(1+o(1))\cdot \sqrt{2\log n}\mathrm{.}$$ We also present candidate maximizers up to dimension $n\le 8$ which appear to be highly structured and have nice closed-form solutions.