Investigations in Combinatorics and Number Theory via Ergodic Theoretic Methods

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Historically, the theory of dynamical systems was rooted in the study of the movement of macroscopic objects (such as moons, planets, etc.) and microscopic particles (such as molecules and atoms). Today, it is a vast area that pursues the study of orbits and transformations in general systems evolving with time. It is a rich and active research area that not only plays an important role in contemporary mathematics, but also greatly contributes to other sciences and finds many applications. This project seeks to improve upon the analytic tools and techniques available in dynamical systems, with anticipated impact in a wide range of fields.The primary objectives of this project are concentrated at the interface of three different areas in pure mathematics: the theory of dynamical systems, combinatorics, and number theory. The employment of analytic tools stemming from measurable, topological, and symbolic dynamics often offers novel ways to analyze number-theoretic and combinatorial situations. This approach has proven to be powerful in solving seemingly intractable problems. In the same vein, this project aims to address, and potentially settle, conjectures and open questions in arithmetic combinatorics and number theory with the help of dynamical methods. The main challenges of the project range from recasting problems in discrete mathematics in a dynamical language to developing the pertinent dynamical tools for solving these reformulations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.