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Multiplicative generalised polynomial sequences

Jakub Konieczny

Abstract

We fully classify completely multiplicative sequences which are given by generalised polynomial formulae, and obtain a similar result for (not necessarily completely) multiplicative sequences under the additional restriction that the sequence is not zero almost everywhere.

Multiplicative generalised polynomial sequences

Abstract

We fully classify completely multiplicative sequences which are given by generalised polynomial formulae, and obtain a similar result for (not necessarily completely) multiplicative sequences under the additional restriction that the sequence is not zero almost everywhere.
Paper Structure (11 sections, 10 theorems, 36 equations)

This paper contains 11 sections, 10 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Background
  3. New results
  4. Open problems
  5. Notation
  6. Background
  7. Generalised polynomials
  8. Nilmanifolds
  9. Representation of generalised polynomials
  10. Equidistribution
  11. Proof of the Main Theorem

Key Result

Theorem A

Let $f \colon \mathbb{N} \to \mathbb{C}$ be a multiplicative generalised polynomial sequence. Then either there exists a periodic multiplicative sequence $\chi \colon \mathbb{N} \to \mathbb{C}$ and an exponent $a \in \mathbb{N}_0$ such that $f(n) = \chi(n)n^a$ for all $n \in \mathbb{N}$, or $f(n) =

Theorems & Definitions (18)

  • Theorem A
  • Remark 1.1
  • Theorem 1.2: ByszewskiKonieczny-2018-TAMS
  • Example 1.3
  • Theorem 1.4: Corollary of Konieczny-2022-JLMS
  • Theorem B
  • proof
  • Theorem 2.1: Corollary of BergelsonLeibman-2007
  • Lemma 3.1
  • Remark 3.2
  • ...and 8 more