Talk #1: Ted Fu, University of Waterloo
"On Waring's problem for large powers"
Let G(k) be the least number s having the property that every sufficiently large natural number is the sum of at most s positive integer k-th powers. In this talk, I will present how Brüdern and Wooley implement smooth numbers technologies in their minor arc analysis and derive G(k) ≤ ⌈k(log k + 4.20032)⌉.
Talk #2: Aidan Boyle, University of Waterloo
"Waring’s problem: Beyond Freiman’s Theorem"
Suppose that we are given a non-decreasing sequence of positive integers (ki) where each term is at least 2. Given a positive integer j, we seek to understand the circumstances in which there exists a positive integer s := s(j) such that every sufficiently large natural number n can be written as a sum of s positive integers to the respective powers kj, ..., kj+s-1. Freĭman asserted that such representation exists if and only if the infinite summation of all 1/ki diverges. We provide an effective version of this theorem, and in particular, comment on instances in which the exponents form a sequence of consecutive terms of an arithmetic progression.
MC 5417