On the structure of bad science matrices

Our main contribution is an explicit construction of an $n \times n$ matrix $A$ showing that $β (A) \geq \sqrt{\log_{2} (n + 1)}$ 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 $n \leq 4$ .

Keywords

discrepancy theory, combinatorics, bad science matrices

Mathematical Subject Classification

Primary: 68R05

Secondary: 52A20, 90C27

Milestones

Received: 7 August 2024

Revised: 3 December 2024

Accepted: 2 January 2025

Published: 12 June 2026

Communicated by Glenn Hurlbert

Authors

Alex Albors
University of Washington Seattle, WA United States

Hisham Bhatti
University of Washington Seattle, WA United States

Lukshya Ganjoo
University of Washington Seattle, WA United States

Raymond Guo
University of Washington Seattle, WA United States

Dmitriy Kunisky
Department of Applied Mathematics and Statistics Johns Hopkins University Baltimore, MD United States

Rohan Mukherjee
University of Washington Seattle, WA United States

Alicia Stepin
University of Washington Seattle, WA United States

Tony Zeng
Department of Mathematics University of Washington Seattle, WA United States

Vol. 19, No. 3, 2026

Alex Albors, Hisham Bhatti, Lukshya Ganjoo, Raymond Guo, Dmitriy Kunisky, Rohan Mukherjee, Alicia Stepin and Tony Zeng

Abstract

Keywords

Mathematical Subject Classification

Milestones

Authors