Analysis Seminar

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Ergodic theorems for arithmetic sets

Speaker: Mariusz MIREK, Rutgers University

Location: Warren Weaver Hall 1302

Date: Thursday, March 5, 2020, 11 a.m.

Synopsis:

The main aim of this talk is to give an introduction to pointwise
ergodic theorems for arithmetic sets. Namely, given $d, k\in\mathbb N$, let
$P_j$ be an integer-valued polynomial of $k$ variables for every
$1\le j\le d$.  Suppose that $(X, \mathcal{B}, \mu)$ is a
$\sigma$-finite measure space with a family of invertible commuting
and measure preserving transformations $T_1, \ldots,T_{d}$ on
$X$. For every $N\in\mathbb N$ and $x\in X$ we define the ergodic
Radon averaging operators by setting
\[    
 A_N f(x)
    = \frac{1}{N^{k}}\sum_{m \in [1, N]^k\cap\mathbb Z^k} 
    f\big(T_1^{ P_1(m)} \circ \ldots
        \circ T_{d}^{ P_{d}(m)} x\big).
\]
We will show that for every $p>1$ and for every function $f\in L^p(X, \mu)$, there is a function 
$f^*\in L^p(X, \mu)$ such that
\[
\lim_{N\to\infty}A_Nf(x)=f^*(x)
  \]
  $\mu$-almost everywhere on $X$. We will achieve this by considering
  $r$-variational estimates.