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Computing the Density of the Positivity Set for Linear Recurrence Sequences

Edon Kelmendi

Abstract

The set of indices that correspond to the positive entries of a sequence of numbers is called its positivity set. In this paper, we study the density of the positivity set of a given linear recurrence sequence, that is the question of how much more frequent are the positive entries compared to the non-positive ones. We show that one can compute this density to arbitrary precision, as well as decide whether it is equal to zero (or one). If the sequence is diagonalisable, we prove that its positivity set is finite if and only if its density is zero. Further, arithmetic properties of densities are treated, in particular we prove that it is decidable whether the density is a rational number, given that the recurrence sequence has at most one pair of dominant complex roots. Finally, we generalise all these results to symbolic orbits of linear dynamical systems, thereby showing that one can decide various properties of such systems, up to a set of density zero.

Computing the Density of the Positivity Set for Linear Recurrence Sequences

Abstract

The set of indices that correspond to the positive entries of a sequence of numbers is called its positivity set. In this paper, we study the density of the positivity set of a given linear recurrence sequence, that is the question of how much more frequent are the positive entries compared to the non-positive ones. We show that one can compute this density to arbitrary precision, as well as decide whether it is equal to zero (or one). If the sequence is diagonalisable, we prove that its positivity set is finite if and only if its density is zero. Further, arithmetic properties of densities are treated, in particular we prove that it is decidable whether the density is a rational number, given that the recurrence sequence has at most one pair of dominant complex roots. Finally, we generalise all these results to symbolic orbits of linear dynamical systems, thereby showing that one can decide various properties of such systems, up to a set of density zero.

Paper Structure

This paper contains 14 sections, 26 theorems, 141 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Sequences and Densities
  3. Strongly Non-Degenerate Subsequences
  4. Period $P_1$
  5. Period $P_2$
  6. Properties of the Subsequences
  7. The Density 1 Problem
  8. Complexity Lower Bounds
  9. The Case of Diagonalisable Sequences
  10. Computing the Density
  11. When is the Density a Rational Number?
  12. Density as a Period
  13. One Pair of Dominant Complex Roots
  14. Toric Words and Linear Dynamical Systems

Key Result

Theorem 1.3

There is a procedure that inputs a lrs and decides whether the density of its positivity set is equal to 1.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (51)

  • Example 1.1
  • Example 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Example 1.6
  • Theorem 1.7
  • Example 2.1
  • Theorem 2.2: bell05_posit_set_recur_sequen
  • Theorem 2.3: Canny_1988 and renegar1992computational, respectively
  • ...and 41 more