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On the number of residues of certain second-order linear recurrences

Federico Accossato, Carlo Sanna

TL;DR

The paper shows that for the second-order family $f = X^2 - a_1 X - 1$ with any nonzero $a_1$, the set of achievable residue counts $ ho$ across all integral recurrences is all positive integers. The authors build this via Lehmer sequences and primitive divisors, relating divisibility of Lehmer terms to multiplicative orders modulo primes, and then carefully craft initial conditions to realize prescribed residue counts modulo suitable moduli. Key contributions include a constructive algorithm to realize any target $n$ as a residue count, and a corollary connecting these constructions to fractional parts of powers, providing control over the number of limit points of $\\operatorname{frac}(\\xi\\alpha^n)$ for certain algebraic $\\alpha$. The methods illuminate the interaction between linear recurrences, primitive divisors, and dynamical properties of algebraic powers, with potential implications for Diophantine and equidistribution-type questions in quadratic fields.

Abstract

For every monic polynomial $f \in \mathbb{Z}[X]$ with $\operatorname{deg}(f) \geq 1$, let $\mathcal{L}(f)$ be the set of all linear recurrences with values in $\mathbb{Z}$ and characteristic polynomial $f$, and let \begin{equation*} \mathcal{R}(f) := \big\{ρ(\mathbf{x}; m) : \mathbf{x} \in \mathcal{L}(f), \, m \in \mathbb{Z}^+ \big\} , \end{equation*} where $ρ(\mathbf{x}; m)$ is the number of distinct residues of $\mathbf{x}$ modulo $m$. Dubickas and Novikas proved that $\mathcal{R}(X^2 - X - 1) = \mathbb{Z}^+$. We generalize this result by showing that $\mathcal{R}(X^2 - a_1 X - 1) = \mathbb{Z}^+$ for every nonzero integer $a_1$. As a corollary, we deduce that for all integers $a_1 \geq 1$ and $k \geq 4$ there exists $ξ\in \mathbb{R}$ such that the sequence of fractional parts $\big(\!\operatorname{frac}(ξα^n)\big)_{n \geq 0}$, where $α:= \big(a_1 + \sqrt{a_1^2 + 4}\,\big) / 2$, has exactly $k$ limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.

On the number of residues of certain second-order linear recurrences

TL;DR

The paper shows that for the second-order family with any nonzero , the set of achievable residue counts across all integral recurrences is all positive integers. The authors build this via Lehmer sequences and primitive divisors, relating divisibility of Lehmer terms to multiplicative orders modulo primes, and then carefully craft initial conditions to realize prescribed residue counts modulo suitable moduli. Key contributions include a constructive algorithm to realize any target as a residue count, and a corollary connecting these constructions to fractional parts of powers, providing control over the number of limit points of for certain algebraic . The methods illuminate the interaction between linear recurrences, primitive divisors, and dynamical properties of algebraic powers, with potential implications for Diophantine and equidistribution-type questions in quadratic fields.

Abstract

For every monic polynomial with , let be the set of all linear recurrences with values in and characteristic polynomial , and let \begin{equation*} \mathcal{R}(f) := \big\{ρ(\mathbf{x}; m) : \mathbf{x} \in \mathcal{L}(f), \, m \in \mathbb{Z}^+ \big\} , \end{equation*} where is the number of distinct residues of modulo . Dubickas and Novikas proved that . We generalize this result by showing that for every nonzero integer . As a corollary, we deduce that for all integers and there exists such that the sequence of fractional parts , where , has exactly limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.
Paper Structure (7 sections, 19 theorems, 36 equations, 1 table)

This paper contains 7 sections, 19 theorems, 36 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lehmer sequences
  4. Multiplicative orders
  5. Second order linear recurrences
  6. Proof of Theorem \ref{['thm:main']}
  7. Proof of Corollary \ref{['cor:fractional-parts']}

Key Result

Theorem 1.1

Let $a_1$ be a nonzero integer. Then $\mathcal{R}(X^2 - a_1 X - 1) = \mathbb{Z}^+$. In other words, for every $n \in \mathbb{Z}^+$ there exist $\bm{x} \in \mathcal{L}(f)$ and $m \in \mathbb{Z}^+$ such that $\rho(\bm{x}; m) = n$. Moreover, one can choose $\bm{x}$ and $m$ so that all the residues of $

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.1
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 29 more