Ergodic theorems for polynomials in nilpotent groups

TL;DR

This paper proves ergodic theorems for polynomials in nilpotent groups, extending classical results to more general algebraic structures and providing new convergence and recurrence theorems in ergodic theory.

Contribution

It establishes the IP polynomial Szemerédi theorem and multiple term return times theorem for nilpotent groups, extending ergodic theorems to locally compact second countable amenable groups.

Findings

Proved IP polynomial Szemerédi theorem for nilpotent groups

Extended convergence theorems for nilpotent polynomial multiple ergodic averages

Generalized return times theorem to locally compact second countable amenable groups

Abstract

The principal results proved in this expository thesis are the IP polynomial Szemer\'edi theorem for nilpotent groups and the multiple term return times theorem with nilsequence weights. It also contains extensions of the convergence theorem for nilpotent polynomial multiple ergodic averages and the return times theorem to locally compact second countable amenable groups.