Quasi-random behavior of deterministic sequences

The goal of this project is to advance the understanding of the local-scale distribution of polynomial sequences modulo 1, and to examine the behavior of various morphic sequences along subsets of integers. The distribution modulo 1 is of great interest in number theory and quantum chaos, particularly in light of the Berry and Tabor conjectures concerning the energy levels of integrable systems. A common random property is equidistribution, which was extensively studied in the classical works of Weyl and others. Recently, Rudnick, Sarnak, and Zaharescu initiated the study of local- scale properties of such sequences, including gap distribution. Currently, there are only a few examples for which the limiting gap distribution is known, with partial progress made in works by Elkies, McMullen, Rudnick, Strömbergsson, Aistleitner, Marklof, and others. The goal of the first part of the project is to approach the gap distribution of quadratic monomials by considering a number of easier problems including smaller degrees and metric properties using various methods from analytic number theory, probability, Diophantine approximations, and homogeneous dynamics. The second part of the project concerns the Sarnak conjecture, which states that any sequence arising from a zero-entropy dynamical system is orthogonal to the Möbius function. Although the conjecture has been verified in numerous specific cases, it remains largely open. One particularly interesting case involves automatic sequences, such as the Thue-Morse sequence, where notable progress has been achieved by Mauduit, Rivat, Drmota, Müllner, and Spiegelhofer. This project aims to extend their results to more complex cases, such as morphic sequences, intersections of multiple automatic sets, and subsequences along higher-degree polynomials.