Nonstandard Convergence Gives Bounds on Jumps
Logic
2017-11-21 v2
Abstract
If we know that some kind of sequence always converges, we can ask how quickly and how uniformly it converges. Many convergent sequences converge non-uniformly and, relatedly, have no computable rate of convergence. However proof-theoretic ideas often guarantee the existence of a uniform "meta-stable" rate of convergence. We show that obtaining a stronger bound---a uniform bound on the number of jumps the sequence makes---is equivalent to being able to strengthen convergence to occur in the nonstandard numbers. We use this to obtain bounds on the number of jumps in nonconventional ergodic averages.
Cite
@article{arxiv.1705.10355,
title = {Nonstandard Convergence Gives Bounds on Jumps},
author = {Henry Towsner},
journal= {arXiv preprint arXiv:1705.10355},
year = {2017}
}