Nonstandard Convergence Gives Bounds on Jumps

Henry Towsner

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arXiv · 1705.10355

Nonstandard Convergence Gives Bounds on Jumps

Henry Towsner

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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.

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BibTeX RIS

Henry Towsner. 2017-05-29. Nonstandard Convergence Gives Bounds on Jumps. https://arxiv.org/abs/1705.10355

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