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On some classification problems of multiplicative functions

S. Kasjan, O. Klurman, M. Lemańczyk

TL;DR

This work classifies bounded multiplicative functions by a dynamical lens, showing that Toeplitz sequences among such functions are exactly those that match a Dirichlet character off a finite set of primes and are pretentious with unique Furstenberg systems. It builds a chain of implications: Toeplitz multiplicative sequences are regular with a single Furstenberg system, automatic multiplicative sequences are governed by a simple structural form, and pretentious functions with a unique Furstenberg system coincide with Besicovitch rational almost periodicity (RAP). The paper also confirms the Frantzikinakis–Host conjecture within the pretentious class and connects this to the corrected Elliott conjecture, while deepening the understanding of GHK semi-norms for pretentious functions and clarifying the landscape of aperiodic subclasses. The results have implications for understanding randomness vs structure in multiplicative functions and relate number-theoretic structure to dynamical properties such as entropy and spectral type.

Abstract

We prove that a multiplicative function $f:\mathbb{N}\to\mathbb{C}$ is Toeplitz if and only if there are a Dirichlet character $χ$ and a finite subset $F$ of prime numbers such that $f(n)=χ(n)$ for each $n$ which is coprime to all numbers from $F$. All such functions bounded by~1 are necessarily pretentious and they have exactly one Furstenberg system. Moreover, we characterize the class of pretentious functions that have precisely one Furstenberg system as those being Besicovitch (rationally) almost periodic. As a consequence, we show that the corrected Elliott's conjecture implies Frantzikinakis-Host's conjecture on the uniqueness of Furstenberg system for all real-valued bounded by~1 multiplicative functions. We also clarify relations between different classes of aperiodic multiplicative functions.

On some classification problems of multiplicative functions

TL;DR

This work classifies bounded multiplicative functions by a dynamical lens, showing that Toeplitz sequences among such functions are exactly those that match a Dirichlet character off a finite set of primes and are pretentious with unique Furstenberg systems. It builds a chain of implications: Toeplitz multiplicative sequences are regular with a single Furstenberg system, automatic multiplicative sequences are governed by a simple structural form, and pretentious functions with a unique Furstenberg system coincide with Besicovitch rational almost periodicity (RAP). The paper also confirms the Frantzikinakis–Host conjecture within the pretentious class and connects this to the corrected Elliott conjecture, while deepening the understanding of GHK semi-norms for pretentious functions and clarifying the landscape of aperiodic subclasses. The results have implications for understanding randomness vs structure in multiplicative functions and relate number-theoretic structure to dynamical properties such as entropy and spectral type.

Abstract

We prove that a multiplicative function is Toeplitz if and only if there are a Dirichlet character and a finite subset of prime numbers such that for each which is coprime to all numbers from . All such functions bounded by~1 are necessarily pretentious and they have exactly one Furstenberg system. Moreover, we characterize the class of pretentious functions that have precisely one Furstenberg system as those being Besicovitch (rationally) almost periodic. As a consequence, we show that the corrected Elliott's conjecture implies Frantzikinakis-Host's conjecture on the uniqueness of Furstenberg system for all real-valued bounded by~1 multiplicative functions. We also clarify relations between different classes of aperiodic multiplicative functions.
Paper Structure (24 sections, 30 theorems, 127 equations)

This paper contains 24 sections, 30 theorems, 127 equations.

Table of Contents

  1. Pretentious and aperiodic multiplicative functions
  2. Main aim of the paper
  3. Pretentious multiplicative functions, Furstenberg systems and Toeplitz sequences
  4. Automatic sequences and multiplicative functions
  5. Precise description of automatic sequences: periodic multiplicative functions and $p$-valuation
  6. Pretentious functions with one Furstenberg system. The Frantzikinakis-Host conjecture
  7. The GHK semi-norms of pretentious functions
  8. Relations between subclasses of aperiodic functions
  9. Toeplitz case -- proof of Theorem \ref{['t:glowne']}
  10. When a pretentious function $f$ has precisely one Furstenberg system?
  11. Motivating examples
  12. Characterization of multiplicative RAP
  13. The FH-conjecture
  14. More on RAP multiplicative functions
  15. GHK semi-norms for pretentious functions
  16. ...and 9 more sections

Key Result

Theorem 1

Let $f:{\mathbb{N}}\to{\mathbb{C}}$ be a nonzero multiplicative function. Then $f$ is Toeplitz if and only if rowf is satisfied, i.e. there are a Dirichlet character $\chi$ and a finite set $F\subset\mathbb{P}$ such that $f(n)=\chi(n)$ for all $n$ coprime with all numbers in $F$. Moreover, each mult

Theorems & Definitions (59)

  • Theorem 1
  • Remark 2
  • Proposition 3
  • Proposition 4
  • Theorem 5
  • Corollary 6
  • Proposition 7
  • Remark 1.1
  • Lemma 1.2
  • proof
  • ...and 49 more