Inverting the Furstenberg correspondence

TL;DR

This paper demonstrates that the Furstenberg correspondence process can be inverted, allowing finite sets to approximate any shift-invariant measure on Cantor space, with explicit bounds and computability implications.

Contribution

It introduces an inversion of the Furstenberg correspondence principle, providing explicit bounds and extending to countable discrete amenable groups.

Findings

Finite sets can approximate shift-invariant measures arbitrarily well.

Every computable invariant measure has a computable generic point.

The inversion applies to countable discrete amenable groups.

Abstract

Given a sequence of subsets A_n of {0,...,n-1}, the Furstenberg correspondence principle provides a shift-invariant measure on Cantor space that encodes combinatorial information about infinitely many of the A_n's. Here it is shown that this process can be inverted, so that for any such measure there are finite sets whose combinatorial properties approximate it arbitarily well. Moreover, we obtain an explicit upper bound on how large n has to be to obtain a sufficiently good approximation. As a consequence of the inversion theorem, we show that every computable invariant measure on Cantor space has a computable generic point. We also present a generalization of the correspondence principle and its inverse to countable discrete amenable groups.