New York Journal of Mathematics
Volume 25 (2019), 651-667


Henry Towsner

Nonstandard convergence gives bounds on jumps

view    print

Published: August 15, 2019.
Keywords: ergodic theorem, metastable convergence, variation bounds.
Subject: 03H05, 37A25.

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.


Partially supported by NSF grant DMS-1600263.

Author information

Henry Towsner:
Department of Mathematics
University of Pennsylvania
209 South 33rd Street
Philadelphia, PA 19104-6395, USA