**Speaker**

Vivian Kuperberg, ETH Zurich

**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 ongoing 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.