On the Structure of Bad Science Matrices
<|reference_start|>On the Structure of Bad Science Matrices: The bad science matrix problem consists in finding, among all matrices $A \in \mathbb{R}^{n \times n}$ with rows having unit $\ell^2$ norm, one that maximizes $\beta(A) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} \|Ax\|_\infty$. Our main contribution is an explicit construction of an $n \times n$ matrix $A$ showing that $\beta(A) \geq \sqrt{\log_2(n+1)}$, which is only 18% smaller than the asymptotic rate. We prove that every entry of any optimal matrix is a square root of a rational number, and we find provably optimal matrices for $n \leq 4$.<|reference_end|>
@article{albors2024on,
    title={On the Structure of Bad Science Matrices},
    author={Alex Albors, Hisham Bhatti, Lukshya Ganjoo, Raymond Guo, Dmitriy
  Kunisky, Rohan Mukherjee, Alicia Stepin, Tony Zeng},
    journal={arXiv preprint arXiv:2408.00933},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.00933},
    primaryClass={math.FA cs.DM math.CO}
}