Geometric discrepancy

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Discrepancy theory, also known as the theory of irregularities of distribution, explores questions such as the "most uniform" way to distribute n points in a unit square and the inherent "irregularity" in any distribution. To address these questions, we quantify irregularity through a numerical parameter known as discrepancy. Initial inquiries began in the 1930s, driven by number theory, and by the late 1960s, a satisfactory solution to the basic discrepancy problem in the plane emerged. However, the higher-dimensional counterpart remains unresolved. Over time, discrepancy theory has evolved into a diverse field, encompassing areas linked to its number theoretic origins, Ramsey theory, hypergraphs, and practical applications in numerical integration and related tasks. Its applications span various domains, including finance, computer graphics, and computational physics. This introductory textbook on discrepancy theory is designed for early graduate students in mathematics or theoretical computer science. Notably, half of the content presents material previously found only in original research papers or surveys, making it a valuable resource for both students and researchers.