Seminar in Algebraic Geometry and Number Theory

Abstract: In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: for an integer $k \ge 2$, consider the set of $k$-tuples of reduced fractions $\frac{a_1}{q_1}, \dots, \frac{a_k}{q_k} \in I$, where $I$ is an interval around $0$. How many $k$-tuples are there with $\sum_i \frac{a_i}{q_i} \in \mathbb Z$?

When $k$ is even, the answer is well-known: the main contribution to the number of solutions comes from “diagonal” terms, where the fractions $\frac{a_i}{q_i}$ cancel in pairs. When $k$ is odd, the answer is much more mysterious! In work with Bloom, we prove a near-optimal upper bound on this problem when $k$ is odd. I will also discuss applications of this problem to estimating moments of the distributions of primes and reduced residues.