Normal numbers and completeness results for difference sets

TL;DR

This paper investigates the complexity of certain sets of real numbers from ergodic theory, establishing their completeness in specific difference hierarchies within descriptive set theory.

Contribution

It demonstrates that natural sets in ergodic theory are complete in the classes of 2-differences and ω-differences of bc^0_3 sets, revealing their precise descriptive complexity.

Findings

Sets are complete in bc^0_3 in the difference hierarchy.

Results establish the exact descriptive set-theoretic complexity of these sets.

The work connects ergodic theory with descriptive set theory through complexity classifications.

Abstract

We consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes $\mathcal D_2 (\mathbf\Pi^0_3)$ and $\mathcal D_\omega (\mathbf \Pi^0_3)$, that is, the class of sets which are 2-differences (respectively, $\omega$-differences) of $\mathbf \Pi^0_3$ sets.