2023 Program for Women and Mathematics: Patterns in Integers: dynamical and number theoretic approaches

Abstract: I will discuss how the Bentkus-Götze-Freeman variant of the Davenport-Heilbronn circle method can be used to study $\mathbb{F}_q[t]$ solutions to inequalities of the form $$\text{ord}(\lambda_1 x_1^k + \cdots + \lambda_s x_s^k - \tau) < \epsilon,$$ where constants $\lambda_1, \ldots, \lambda_s, \tau \in \mathbb{F}_q((1/t))$ satisfy certain conditions.  After introducing some function field notation, I will give a sketch of how to count $\mathbb{F}_q[t]$ solutions in this setting.