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On (almost) realizable subsequences of linearly recurrent sequences

Florian Luca, Tom Ward

Abstract

In this note we show that if $(u_n)_{n\geqslant 1}$ is a simple linearly recurrent sequence of integers whose minimal recurrence of order $k$ involves only positive coefficients that has positive initial terms, then $(Mu_{n^s})_{n\geqslant 1}$ is the sequence of periodic point counts for some map for a suitable positive integer $M$ and $s$ any sufficiently large multiple of $k!$. This extends a result of Moss and Ward [The Fibonacci Quarterly 60 (2022), 40-47] who proved the result for the Fibonacci sequence.

On (almost) realizable subsequences of linearly recurrent sequences

Abstract

In this note we show that if is a simple linearly recurrent sequence of integers whose minimal recurrence of order involves only positive coefficients that has positive initial terms, then is the sequence of periodic point counts for some map for a suitable positive integer and any sufficiently large multiple of . This extends a result of Moss and Ward [The Fibonacci Quarterly 60 (2022), 40-47] who proved the result for the Fibonacci sequence.
Paper Structure (11 sections, 3 theorems, 27 equations)

This paper contains 11 sections, 3 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['thm:1']}
  3. The algebraic Dold condition
  4. $\pi\vert\lambda$
  5. $p\!\nmid\!\lambda$
  6. The sign condition
  7. The proof of Theorem \ref{['thm:2']}
  8. The proof of Theorem \ref{['thm:3']}
  9. The case $Q=0$
  10. The case $(P,Q)=(0,1)$
  11. Remaining cases

Key Result

Theorem 1

Assume that $F$ has only simple zeros.

Theorems & Definitions (5)

  • Theorem 1
  • Remark
  • Theorem 2
  • Conjecture 1
  • Theorem 3