This volume includes a collection of eight articles in number theory, dedicated to research topics explored during the 2020/2021 Jean Morlet Chair program, titled Diophantine Problems: Determinism, Randomness, and Applications. The volume contains both review articles and original contributions in Diophantine number theory and its applications.

The most detailed contribution focuses on equidistribution and discrepancy theory from a probabilistic perspective. In this article, Aistleitner, Berkes, and Tichy study lacunary trigonometric sums and lacunary sums of dilated functions. By employing a combination of tools (probabilistic methods, Diophantine geometry methods, etc.), they significantly generalize several classical results by Salem, Zygmund, Erdős, Gal, and others in Fourier analysis and metric number theory. Two articles, one by Kreso and Tichy and the other by Heintze, focus on polynomial variants of the Diophantine equations of Pillai, which have been widely studied since the 1930s. Both articles present new results and use classical results on null sums in function fields to establish analogues of Pillai’s asymptotic results in the context of polynomials and sums of polynomial powers. Furthermore, the article by Kreso and Tichy makes use of Ritt’s polynomial decomposition theory, which has broad applications in number theory, complex analysis, and beyond.