AN APPROACH TO POINTWISE ERGODIC THEOREMS
BEN KRAUSE
1. Abstract
The study of pointwise ergodic theorems goes back to 1931, when Birkhoff proved his classical pointwise ergodic theorem
for Cesaro averages:
Theorem 1 (Special Case). For any non-atomic probability space, (X, µ), and T : X → X a measure-preserving transformation, the means


1 X

T n f (x)
t

n≤t
2
converge pointwise µ-a.e. for each f ∈ L (X).
It took almost 60 years before the linear averages could be replaced by more complicated polynomial ones; in the late
eighties, Bourgain managed to generalize Birkhoff’s theorem as follows:
Theorem 2 (Special Case). For any non-atomic probability space, (X, µ), and T : X → X a measure-preserving transformation, the means



1 X 2
T n f (x)

t
n≤t
2
converge pointwise µ-a.e. for each f ∈ L (X).
Although his theorem is very abstract, Bourgain achieved his result through (surprisingly) hard-analytic technique: he
used the analytic number-theoretic circle method of Hardy and Littlewood to gain quantitative control over the oscillations
of the pertaining Cesaro means. The ideas used were very robust! – they can be actually be adapted to quantify the rate
at which convergence of the square-means occurs.
In this talk we will introduce the quantitative pointwise ergodic machinery first in the context of Birkhoff’s theorem,
and then apply it to the case of the square means.
UCLA Math Sciences Building, Los Angeles, CA 90095-1555, USA
E-mail address: benkrause23@math.ucla.edu
Date: October 7, 2014.
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