Félix Brokering Pinilla
University of Bristol
Title
Gowers Uniformity and Pointwise (Non-)Convergence of Weighted Ergodic Averages
Abstract
In this talk, we will discuss joint work with Ben Krause on pointwise convergence of weighted ergodic averages. After linking pointwise convergence to quantitative decay in Gowers norms, we present our main result: there exist 1-bounded weights whose Gowers norms all decay to zero, and yet whose weighted averages fail to converge pointwise on some measure preserving system. We also show that this phenomenon is generic in a Baire sense and, as an application, obtain arbitrarily dense Gowers uniform subsets of the natural numbers along which pointwise convergence fails. Time permitting, we will go over some of the ideas behind the construction of a weight whose decay in Gowers norms comes arbitrarily close to the threshold ensuring pointwise convergence, and yet whose averages diverge almost everywhere, showing that this threshold is, in a sense, sharp.
