Ben Krause: “Discrete Analogues in Harmonic Analysis”. Graduate Studies in Mathematics 224

The article under review provides a comprehensive introduction to the field of discrete analogues in harmonic analysis (DAHA), a subject that has seen significant developments since the late 1980s. The origins of DAHA are traced back to the intersection of number theory and analysis, with a focus on the circle method and its applications. The review discusses the key problems and seminal results in DAHA, highlighting the contributions of Arkhipov–Oskolkov, Bourgain, and Stein–Wainger. The linear theory in DAHA is explored in detail, with a focus on linear ergodic averages and recent results in pointwise bilinear and multilinear ergodic theorems. The review also provides a historical context for the development of DAHA, from its antecedents in number theory and analysis to its current status as a field in its own right. The circle method is identified as a crucial tool in the analysis of discrete analogues, with its low versus high frequency analysis providing a deeper analogy between continuous and discrete results. The review concludes with a discussion of the book's exposition, highlighting its excellent choice and presentation of topics, and its focus on motivating proofs and describing unifying themes.