 

Henry Towsner
Nonstandard convergence gives bounds on jumps
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Published: 
August 15, 2019. 
Keywords: 
ergodic theorem, metastable convergence, variation bounds. 
Subject: 
03H05, 37A25. 


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 nonuniformly and, relatedly, have no computable rate of convergence. However
prooftheoretic ideas often guarantee the existence of a uniform ``metastable'' rate of convergence.
We show that obtaining a stronger bounda uniform bound on the number of jumps the sequence makesis 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. 

Acknowledgements
Partially supported by NSF grant DMS1600263.


Author information
Henry Towsner:
Department of Mathematics
University of Pennsylvania
209 South 33rd Street
Philadelphia, PA 191046395, USA
htowsner@math.upenn.edu

