Nonstandard Convergence Gives Bounds on Jumps

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.

References in corpus (2)

• Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees
• Model theory and metric convergence I: Metastability and dominated convergence

Cited by in corpus (1)

• Fluctuation bounds for ergodic averages of amenable groups on uniformly convex Banach spaces