Explore the application of discrepancy theory to enhance randomized controlled trial design in this comprehensive lecture. Delve into the potential outcomes model, experimental design, and average treatment effect. Examine the Thompson estimator, difference of means, and methods for measuring variance. Investigate balanced designs, covariance, and the Gram-Schmidt walk algorithm. Analyze sub-Gaussian tails, algorithm explanations, and projections. Gain insights into variance calculations, ideal cases, and phases of the process. Discover how to achieve better confidence intervals and understand the crucial factors influencing trial outcomes. Learn about orthonormal bases and the potential for improvement in randomized controlled trial methodologies.