Ergodic bounded constructions, Disjointness and Weak Limits of Powers

TL;DR

This paper provides simplified proofs of key theorems related to ergodic bounded constructions, including Bourgain's theorem on Mobius orthogonality and properties of bounded-recurrent constructions with Markov self-similarity.

Contribution

It offers a concise proof of Bourgain's Mobius orthogonality theorem for bounded rank-one constructions and simplifies the proof regarding flat-recurrence of bounded-recurrent constructions with Markov self-similarity.

Findings

Proof of Bourgain's theorem on Mobius orthogonality for bounded rank-one constructions.

Simplified demonstration that bounded-recurrent constructions with Markov self-similarity are flat-recurrent.

Enhanced understanding of ergodic properties related to disjointness and weak limits of powers.

Abstract

In connection with some recent results by J. Bourgain and H. Abdalaoui, M. Lemanczyk, T. De La Rue (ALR) we present a short proof of Bourgain's theorem on Mobius orthogonality property for bounded rank-one constructions. The proof of the fact (due to ALR) that bounded-recurrent constructions with Markov self-similarity have to be flat-recurrent is simplified as well.(v4 in Russian)