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A note on Sarnak processes

Mariusz Lemańczyk, Michal D. Lemańczyk, Thierry de La Rue

Abstract

Basic properties of stationary processes called Sarnak processes are studied. As an application, a combinatorial reformulation of Sarnak's conjecture on M{ö}bius orthogonality is provided.

A note on Sarnak processes

Abstract

Basic properties of stationary processes called Sarnak processes are studied. As an application, a combinatorial reformulation of Sarnak's conjecture on M{ö}bius orthogonality is provided.
Paper Structure (17 sections, 1 theorem, 63 equations)

This paper contains 17 sections, 1 theorem, 63 equations.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Results
  4. Main results
  5. Motivation
  6. Chowla conjecture
  7. Why Sarnak processes are interesting from the analytic number theory point of view?
  8. Examples of Sarnak processes
  9. Proofs
  10. Background for \ref{['theo:p:alaweakB']}
  11. Proof of \ref{['theo:p:alaweakB']}
  12. Background for \ref{['theo:t:mnozenie']}
  13. Proof of \ref{['theo:t:mnozenie']}
  14. Background for \ref{['theo:p:thierry']}
  15. Proof of \ref{['theo:p:thierry']}
  16. ...and 2 more sections

Key Result

Theorem 5.1

Let $(Y,\mathcal{C},\nu,S)$ be a dynamical system and $(Z,\mathcal{D},\kappa,R)$ has entropy zero. Assume that $\rho\in J(S,R)$ is a joining of $\nu$ and $\kappa$. Then i.e. each such joining has to be the relatively independent extension of its restriction to $(Y/\Pi(S))\times Z$.

Theorems & Definitions (13)

  • proof : Proof of the statement from \ref{['exam:e:p2']}
  • proof
  • proof
  • proof : Proof of \ref{['lemm:l:mich']}
  • proof
  • proof
  • proof : Proof of \ref{['coro:t:main']}
  • Theorem 5.1
  • proof
  • proof
  • ...and 3 more