Panoramas et synthèses, n° 62. Diophantine problems : determinism, randomness and applications

This volume comprises a collection of eight papers in number theory, which are devoted to research topics explored during Jean Morlet Chair Program 2020/2021, titled Diophantine problems: Determinism, Randomness, and Applications. The volume includes survey articles as well as some original papers in diophantine number theory and applications. The most extensive contribution deals with equidistribution and discrepancy theory from a probalistic point of view. In this article, Aistleitner, Berkes, and Tichy investigate lacunary trigonometric sums and lacunary sums of dilated functions. Using a combination of tools (probabilistic method, methods from Diophantine analysis, etc.) they widely generalize some classical results of Salem, Zygmund, Erdős, Gál and others in Fourier analysis and metric number theory. Two papers, one by Kreso and Tichy and the other by Heintze, focus on polynomial variants of Pillai's Diophantine equations, which have been extensively studied since the 1930s. Both papers present new results and utilize classical findings on vanishing sums in function fields to establish analogues of Pillai's asymptotic results in the context of polynomials and polynomial power sums. The paper of Kreso and Tichy further employs Ritt's theory of polynomial decomposition, which finds broad applications in number theory, complex analysis, etc. Pakovich's contribution revolves around the theory of Ritt and its generalizations, focusing on certain problems and conjectures concerning semigroups of rational functions under the operation of functional composition. The paper by Drmota, Lemańczyk, Müllner and Rivat provides a survey of recent advancements on the Sarnak conjecture within the context of product spaces, prime number theorems, polynomial subsequences, and morphic sequences. It also introduces novel results concerning products of automatic sequences. Rossi, Steiner, and Thuswaldner's paper opens up new directions in the study of arithmetic dynamical systems and their interaction with the analysis of fractal structures. The survey paper by Ökten offers an introduction into quasi-random numbers, pseudorandomness and number-theoretic algorithms for generating pseudorandom sequences, as well as their applications in numerical integration, such as the quasi-Monte Carlo method in mathematical finance. This article has been successfully used by students in summer schools as a first step to get familiar with quasi-Monte Carlo methods, their number theoretic origin and some applications. Finally, a related paper by Nachbagauer and Thonhauser examines an option pricing problem in mathematical finance from the perspective of quasi-Monte Carlo integration.