Nonstandard Convergence Gives Bounds on Jumps

TL;DR

This paper explores the relationship between nonstandard convergence and bounds on the number of jumps in sequences, providing new insights into convergence rates and ergodic averages using proof-theoretic methods.

Contribution

It establishes an equivalence between bounds on jumps in sequences and nonstandard convergence, applying this to ergodic averages.

Findings

Bound on jumps is equivalent to nonstandard convergence.

Proof-theoretic methods can derive bounds on jumps.

Results apply to nonconventional ergodic averages.

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.