Computing the Density of the Positivity Set for Linear Recurrence Sequences

TL;DR

This paper investigates the density of positive entries in linear recurrence sequences, providing methods to compute, decide, and analyze their positivity sets and extending results to linear dynamical systems.

Contribution

It introduces algorithms to compute the positivity set density, decide when it is zero or one, and generalizes these results to linear dynamical systems.

Findings

Density can be computed to arbitrary precision.

Decidability of positivity set density being zero or one.

Extension of results to symbolic orbits of linear systems.

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.